
Citation 
 Permanent Link:
 https://ufdc.ufl.edu/UF00099467/00001
Material Information
 Title:
 Twoparameter stochastic processes with finite variation
 Creator:
 Lindsey, Charles, 1962 ( Dissertant )
Dinouleanu, Nicolae ( Thesis advisor )
Brooks, James ( Reviewer )
Block, Louis ( Reviewer )
Glover, Joseph ( Reviewer )
Keesling, James ( Reviewer )
Dolbier, William ( Reviewer )
Lockhart, Madelyn ( Degree grantor )
 Place of Publication:
 Gainesville, Fla.
 Publisher:
 University of Florida
 Publication Date:
 1988
 Copyright Date:
 1988
 Language:
 English
 Physical Description:
 iii, 152 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Banach space ( jstor )
Martingales ( jstor ) Mathematics ( jstor ) Perceptron convergence procedure ( jstor ) Property lines ( jstor ) Property rights ( jstor ) Quadrants ( jstor ) Rectangles ( jstor ) Topological theorems ( jstor ) Topology ( jstor ) Dissertations, Academic  Mathematics  UF ( local ) Mathematics thesis Ph. D ( local ) Stochastic processes ( lcsh )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Abstract:
 Let E be a Banach space with norm â€¢, and f: R2+ â†’E a function
with finite variation. Properties of the variation are studied, and
an associated increasing realvalued function f is defined.
Sufficient conditions are given for f to have properties analogous to
those of functions of one variable. A correspondence f â†”Î¼f
between
such functions and Evalued Borel measures on R2+ is established, and
the equality  Î¼f = Î¼f is proved. Correspondences between Evalued
twoparameter processes X with finite variation x and Evalued
stochastic measures with finite variation are established. The case
where X takes values in L(E,F) (F a Banach space) is studied, and it
is shown that the associated measure Î¼x takes values in L(E,F"); some
x
sufficient conditions for y to be L(E,F)valued are given. Similar
results for the converse problem are established, and some conditions
sufficient for the equality  Î¼x = Î¼x are given.
 Thesis:
 Thesis (Ph. D.)University of Florida, 1988.
 Bibliography:
 Includes bibliographical references.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Charles Lindsey.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 023644331 ( alephbibnum )
19299436 ( oclc ) AFH3397 ( notis )

Downloads 
This item has the following downloads:

Full Text 
TWOPARAMETER STOCHASTIC PROCESSES
WITH FINITE VARIATION
BY
CHARLES LINDSEY
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
TABLE OF CONTENTS
Page
ABSTRACT .................................... ...................... iii
CHAPTERS
I INTRODUCTION...............................................1
1.1 Notation ............................................. 3
1.2 Filtrations........................................... 4
1.3 Stochastic Processes..................................8
1.4 Stopping Points and Lines.............................8
1.5 Some Measure Theory..................................12
II VECTORVALUED FUNCTIONS WITH FINITE VARIATION.............21
2.1 Basic Definitions and Some Examples..................21
2.2 The Variation of a Function of Two Variables.........26
2.3 Functions of Two Variables With Finite Variation..... 43
III STIELTJES MEASURES ON THE PLANE...........................69
3.1 Measures Associated With Functions...................69
3.2 Functions Associated With Measures...................85
IV VECTORVALUED PROCESSES WITH FINITE VARIATION.............95
4.1 Definitions and Preliminaries........................96
4.2 Measures Associated With VectorValued
Stochastic Functions................................112
4.3 VectorValued Stochastic Functions Associated
With Measures .................................... 131
4.4 On the Equality Iml P= I ......................... 144
V CONCLUSION............................................. 149
BIBLIOGRAPHY....................................................... 150
BIOGRAPHICAL SKETCH.............................. ........... 154
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Patial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
TWOPARAMETER STOCHASTIC PROCESSES
WITH FINITE VARIATION
BY
CHARLES LINDSEY
April, 1988
Chair: Nicolae Dinculeanu
Major Department: Mathematics
2
Let E be a Banach space with norm I1, and f: R + E a function
with finite variation. Properties of the variation are studied, and
an associated increasing realvalued function IfJ is defined.
Sufficient conditions are given for f to have properties analogous to
those of functions of one variable. A correspondence f + pf between
such functions and Evalued Borel measures on R is established, and
the equality uf = ifl is proved. Correspondences between Evalued
twoparameter processes X with finite variation IXI and Evalued
stochastic measures with finite variation are established. The case
where X takes values In L(E,F) (F a Banach space) is studied, and it
is shown that the associated measure p takes values in L(E,F"); some
sufficient conditions for px to be L(E,F)valued are given. Similar
results for the converse problem are established, and some conditions
sufficient for the equality \ux = pUXI are given.
iii
CHAPTER 1
INTRODUCTION
Families of random variables indexed by directed sets have been
objects of study for some time. The most common by far, though, has
2 2
been R and especially the first quadrant of the plane, as this
is considered the most "natural" extension of the usual indexing set R
or R One of the principal objects of study relating to such
processes has been the stochastic integral of processes indexed by the
plane.
Stochastic integration with respect to twoparameter Brownian
motion has been studied extensively; the focus of more recent study
has been the more general problem of extending the oneparameter
theory of stochastic integration with respect to a semimartingale. In
one parameter, this is done by writing a semimartingale X as
X = M + A
with M a locally square integrable local martingale and A a process of
finite variation (see, for example, Dellacherie and Meyer [5]). The
major problem in two parameters is with the notion of "local martingale":
the theory of stopping is not sufficiently welldeveloped to permit a
definition with all the necessary properties. Some preliminary work
has been done, however, using the notion of an increasing path (see,
for example, Fouque [9] and Walsh [18]). A definition of a stopped
process has also been given for square integrable martingales in Meyer
[12], but this too is rather limited.
Another area that has seen renewed interest is the study of
processes with values in a vector space, especially in a Banach space
(see, for example, Neveu [14]). In Dinculeanu [7] the correspondence
given in Dellacherie and Meyer [5] between processes with finite
variation and stochastic measures is extended to the case where the
processes have values in a Banach space, and in Meyer [12] the
correspondence is stated for twoparameter realvalued processes. In
the oneparameter theory, this result finds its use in applications to
projections, what in turn (at least in the scalar case) is relevant to
the decomposition of supermartingales and to semimartingales. This
correspondence X ix is given by
(1.1.1) p (D) = E(t ddXz)
for scalarvalued, bounded process. We shall extend this
correspondence to the case where X has integrable variation, with
values in a Banach space E.
Since the inner integral on the right side of (1.1.1) is computed
pathwise, we begin by studying the properties of Banachvalued
2
functions f: R E with bounded variation. In Chapter II, we
develop the relevant properties of such functions, and in Chapter III
2
we show that to each such f we can associate a measure uf: B(R ) E
with finite variation, and we prove the equality
Il( fI' V flr
(1 .1.2)
(i.e., the variation of the measure associated with f is equal to the
measure associated with the variation of f). In Chapter IV we prove
that a stochastic measure can be associated with a process of
integrable variation, in the same manner as in Dinculeanu [7], and we
consider the converse problem: that of associating a function with a
given stochastic measure. Unfortunately, the equality (1.1.2) is not
preserved in general for processes and stochastic measures, so we end
up by establishing some sufficient conditions for the equality to hold.
1.1 Notation
2
The index set is R ; we shall sometimes consider functions and
processes extended by zero to all of R In the rare cases where we
look at points outside the first quadrant, we shall say so
explicitly. We denote points in R2 by z, u, w, v and their
coordinates by the letters r, s, t, p. For example, then, we write
z = (s,t), u (p,r), etc.
There are two notations commonly used in the literature for the
order relation in R2: we adopt here the notation of Meyer [12]. For
two points z = (s,t), z' = (s',t'), we have z.z' iff sSs', tSt';
z
z#z'.)
We denote by (z,z'] the set of all u such that z
define analogously the rectangles [z,z'], [z,z'), and (z,z'). We
* In their pioneering paper, Cairoli and Walsh [2] use "<" in place
of 5, and "<<" for the strict inequality. This is used some in
later places (e.g., in Walsh [18]), but the notation we adopt seems
to be more common now.
denote [z,o) = {u: zsuj, and by R the rectangle EO,z] (or (,z] if
we are discussing all of R2; these cases will be stated explicitly
when they occur). We shall also have need to state the coordinates of
rectangles explicitly: we write, for example, (z,z'] = ((s,t),(s',t')]
( = l(p,r): s
For a function f defined on R (or R ), and a rectangle
R = (z,z'3 = ((s,t),(s',t')], we define the (rectangular) increment of
f on R, denoted A ,(f) by
zz
A .(f) f(s',t') f(s',t) f(s,t') + f(s,t).
The notation A ,(f) is used for this sum regardless of whether the
zz
rectangle is open or closed.
1.2 Filtrations
Let (n,F,P) be a complete probability space. There are two
different methods in the literature for constructing filtrations
2
on R ; we shall give both.
The first method is the one adopted by Meyer [12]. We begin with
two filtrations satisfying the usual conditions:
(F)sR also denoted (Fs), and
(F2)tR also denoted (F ).
2 1 2
When we wish to discuss all of R we extend these by taking F, Ft
to be the degenerate ofield for s<0 or t
1 2 1 1
convention F = F = F (which may or may not equal F = VF or
F2 = VF2)
t 1 2
We then set Ft = F F and verify that this family satisfies
st s
the usual conditions.
1 2
1) Each Fst contains the Pnegligible sets, since Fs, F contain
them for all s,t.
2) If (s,t) S (s',t'), then Ft C F t..
1 1 2 2
In fact, F1 C F since sSs', F2 C F, since t5t', hence
s s t t
1 2 1 F 2
Fst F ( F Fl C F OF Fs
st s t S t st
3) We have F = ( F ,. (Note: In two parameters,
st st
(s,t)<(s',t')
our usual definition of right limit is for (s,t) S (s',t'),
(s,t) (s',t'), as we shall see later. The statement we
shall prove is somewhat stronger.)
Proof. One containment is evident: since FstC Fs t, for
each (s',t') > (s,t) by (2), Fs C f Fst
t (s,t)<(s',t')
For the other containment, let
A c n F = (F nFi).
(s,t)<(s',t') s'>s
t'>t
Then A E F for all s'>s, hence A E Sn F = F
SS '>s
Similarly, A c F2 for all t'>t, hence A E } F2 = F2
t
Then A E (F1 (F) =2 Ft, hence F ( ( F.).
s t stt st s, s t
s'>s
t'>t
Putting the two containments together gives the equality.
We say that the condition of commutation is satisfied if the
conditional expectation operators E(IFS ) and E(.F ) commute,
s t
i.e., if E(.IF1F ) = E(.IF IF ). The product is then equal to
E(IF1 nF2) = E(IFt). (In fact, denote, for f c L (P), E a Banach
space with the RNP, g = E(flF IF2) = E(fF2 IF1). Then g is measurable
2 1
with respect to F2 and Fs, hence g is F tmeasurable. Also, for
t s st
1 2
A E Fst, we have IAgdP = JAE(f Fsl F)dP = AE(fFs)dP = JAfdP
since A E F = F1 (1 2 Thus, g = E(fIF ).) All of the main
st s t St
results of the theory of twoparameter processes require the condition
of commutation.
The second (and more frequently used) way of describing
filtrations on R2 is due to Cairoli and Walsh [2]. There, we begin
with a family (Fz, z c R2} of subafields of F satisfying the
following axioms:
(Fl) If ziz', then F C F' (this is (2) of Meyer).
z z
(F2) F contains all the negligible sets of F. (Note: This,
o
along with (F1), implies condition (1) of the Meyer
construction.)
(F3) For each z, F = ( F .. (This was condition (3) of
z z'>z z
z'>z
Meyer.)
We now define F = F = V Fst F2 F = V Fst. In place of
t s
the condition of commutation, we impose the axiom
(F4) For each z, F and F are conditionally independent given
z z
F, i.e., for Y F1measurable, integrable, Y2 F2
measurable, integrable, we have
(1.2.1) E(Y1Y2 Fz) = E(YIIF )E(Y21Fz).
This condition is equivalent [4, II.45] to
(1.2.2) E(Y1IF ) = E(Y IFz)
for every F measurable, Integrable r.v. Y1. The condition (F4) can
z
be seen to be equivalent to the condition of commutation as follows:
(F4) => commutation: Let X be Fmeasurable, integrable. Then
E(XIF1) Is F1measurable, integrable. By (F4), E(E(XIF1)IF2)
E(E(XIFz)IFz) = E(XIFz). Similarly, E(XIF2) is F2measurable,
integrable; hence by (F4) E(E(X F2) F) = E(E(X F2)I F) E(XIFz).
Thus E(E(XF1 )IF2) = E(XIFz) = E(E(X F2) Fz), i.e., the operators
E(.IF2) and E(.IF1) commute.
Commutation => (F4): We shall assume commutation and prove that
(1.2.2) holds.
Let Y be F measurable, integrable. We have, from the
1 z
commutation condition,
E(Y jF) = E(YjFz F2) 2 E(Y1 2 )
which Is just (1.2.2).
Although the difference between the two constructions is slight
(the main difference being on the "border at infinity," where we do
not necessarily have right continuity of the filtrations (F ) and
(F ) in the CairoliWalsh model), the condition most often imposed on
t
the filtration in the literature is stated as the (F4) condition,
although the notation used is more often that of Meyer.
1.3 Stochastic Processes
Throughout this section, (0,F,P) is a complete probability space,
(F ) a filtration satisfying axioms (F1)(F4) (in particular, one
z 2
zeR
constructed by the method of Meyer), and E will denote a Banach space
equipped with its Borel ofield, denoted B(E). The definitions in
this section are taken from [12], [7].
Definition 1.3.1. A stochastic process Is a measurable (i.e., a
(B(R )xF, B(E))measurable) function X: R2xn E. We usually denote
X(z,w) by Xz(w), and the mappings w X (w) by Xz
Remark. It will sometimes be convenient to extend the index set to
all of R2 by defining X 0 for z outside the first quadrant, and
F the degenerate ofield for those z. When we wish to consider a
process in this light, we shall say so explicitly.
A brut or raw process is a process X such that X is Fmeasurable
for all z E R X is called adapted if X is F measurable for all
z E R2.
2
A process X is called progressive If, for every z E R+, the map
(z,w) Xz(w) from [0,z]xQ E is (B([0,z])xFz, B(E))measurable.
(Note: This definition comes from Fouque [9]; in Meyer [12]
progressivity is defined using halfopen rectangles [0,z). The
definition we give here is the one in common use today.)
1.4 Stopping Points and Lines
1.4.1 Stopping Points
The notion of stopping plays a fundamental role in the theory of
oneparameter processes. The obvious extension of the definition of
stopping time to twoparameter processes yields what is known as a
stopping point.
Definition 1.4.1. A stopping point is a mapping Z: n R2 such that
2
for all v E R, the set Jz < v} belongs to F .
We often denote Z = (S,T). The components S,T are, It turns out,
stopping times with respect to (F ), (F ), respectively, but this by
s t
itself is insufficient to guarantee that (S,T) Is a stopping point.
We do, however, have the following characterization [12]:
Theorem 1.4.2. Let S be an (F )stopping time, T an (F2)stopping
5 t
2
time. Then Z = (S,T) is a stopping point if and only if S is F
measurable, T F measurable.
Although stopping points have found some application recently
(see, for example Walsh [18] and Fouque [9]), they are rather
inadequate for the purpose of developing a theory of localization for
twoparameter processes. To begin with, due to the partial order
in R2, we are not even assured that jZ>v} F Also, if U and V are
stopping points, UV may not be. Moreover, in one dimension, the graph
of a stopping time T divides R+ x into two components, namely the
stochastic intervals [[O,T)) and [[T,)), whereas the graph of a
stopping point Z does no such thing. Furthermore, an important
realization of a stopping time is as the debut of a progressive set,
and we have no analogous result in the plane.
1.4.2 Stopping Lines
Let AC R 2x be a random set (the usual definition: 1A is a
measurable process). The open envelope of A, denoted (A,), is the
random set whose section for each w c 0 is given by
(A,)(w) = ) (z,).
zeA(w)
Some properties of (A,) (proofs can be found in Meyer [12]):
1) If A is progressive, (A,t) is predictable.
2) The interior of a progressive set A is progressive, from which
we obtain, by passing to complements, that the closure of A is
progressive.
We designate in particular by [A,=) the closure of (A,) (i.e., for
each w, we define [A,")(w) = (A,w)(w)). [A,) is progressive If A is
progressive, by the above properties; it is called the closed envelope
of A. The random set
DA [A,)\ (A,)
is called the debut of A: it is progressive if A is progressive.
Definition 1.4.3. a) A random set Z is called a stopping line If it
is the debut of a progressive set, i.e., if there is a progressive set
A such that
Z = DA = [A,)\ (A,m).
b) A stopping line Z is predictable if it belongs (as a set) to the
predictable ofield P.
Remarks.
1) The set A in the definition is not unique: for example, A
and (A,) always have the same debut.
2) This definition has drawbacks: for example, DA is not
necessarily adherent to A.
3) For an alternate way of defining stopping lines (as a map from
Q into a certain set of curves on R ), see Nualart and Sanz
[15].
4) Since (A,) is predictable, to say that a stopping line
Z = DA is predictable amounts to saying that [A,m) is
predictable, which is an alternate way of defining predictable
stopping times in one parameter (cf. [4, IV.693).
We introduce a partial order on the set of stopping lines by defining
H S K if (H,m) 3 (K,m).
We also make the convention
H < K if (H,m) [K,w).
The set of stopping lines is then a lattice for 5:
HK is the debut of (H,) U (K,m), and
HK is the debut of (H,) l(K,).
We say that a stopping line H is the limit of an increasing (resp.
decreasing) sequence (H ) of stopping lines if [H,m) = th[H ,=)
n
(resp. if (H,m) = J(Hn,')). In the first case, if we have
n
[H,=) = \'(Hn',), we say that the sequence (H n) announces (foretells)
H; if there exists a sequence (H) announcing H, we say H is
n
announceable (foretellable).
In the oneparameter theory, this is equivalent to being
predictable (and in fact many authors define predictability in this
fashion). However, we only have one implication for stopping lines,
namely, that every announceable stopping line is predictable. In
fact, if H is announceable, then ex. (H ) such that [H,) ( ,n'
nn
Each (Hn,.) is predictable, hence the intersection is predictable. By
remark (4), this implies that H is predictable. Unfortunately, the
other implication does not hold. For a counterexample, see Bakry l[].
Finally, we note the existence of a predictable crosssection
theorem for stopping lines. The proof is essentially the same as for
the onedimensional case. We denote by i the projection
of R x onto a.
Theorem 1.4.4 (predictable crosssection theorem). Let A be a
predictable set, and let e>0. There exists a compact* predictable set
K satisfying the following:
1) KC A and P{K = 0, A 0} < E
2) DK is announceable and K C D.
1.5 Some Measure Theory
In this section we collect some results from measure theory which
we shall make frequent use of later.
* That is, the section K(w) is compact for each w.
5.1 Monotone Class Theorems
There are several versions of monotone class theorems, both for
families of sets and for functions. The two stated here are the
variants we shall use later. The statements are taken from
Dellacherie and Meyer [4].
Theorem 1.5.1. Let C be a ring of subsets of Q. Then the monotone
class M generated oy C is equal to the oring generated by C. If C is
an algebra, then M = o(C).
Theorem 1.5.2. Let H be a vector space of bounded realvalued
functions defined on Q, which contains the constants, is closed under
uniform convergence, and has the following property:
for every uniformly bounded increasing sequence (f ) of
positive functions from H, the function f = lim fn belongs
n
to H.
Let C be a subset of H which is closed under multiplication. The
space H then contains all bounded functions measurable with respect to
the afield o(C).
The most frequent application of this theorem comes when we wish
to prove that a certain property holds for all bounded Fmeasurable
functions; it allows us to reduce to the case where f is the indicator
of a set. We shall see numerous examples of this theorem at work.
1.5.2 Liftings
Let (T,E,p) be a measure space, p0. Material for this section
comes from Dinculeanu [6, pp. 199216]. We shall need these
properties in Chapter IV.
Definitions 1.5.3.* Let u be a positive measure.
A mapping p: L(uv) L (p) is said to be a lifting of L()
if it satisfies the following six conditions:
1) p(f) = f pa.e.
2) f = g Ua.e. implies p(f) = p(g)
3) p(af + Bg) = ap(f) + Bp(g) for a,B E R
4) frO implies p(f)W O
5) f(z) = a implies p(f)(z) = a
6) p(fg) = p(f)p(g).
We say that L (u) has the lifting property if there exists a lifting p
of L(p5). The following theorem affirms that, for probability
measures P in particular, L (P) always has the lifting property.
Theorem 1.5.4. If i has the direct sum property, then L (u) has the
lifting property.
Let, now, E and F be Banach spaces, T a set, and Z a subspace
of F', norming for F, i.e., such that
lYIF = sup i for every ycF.
ZEZ II"F'
For every function U: T L(E,F) (continuous linear maps E F),
x: T 4 E and z: T + Z, we denote by the map t
and by [JU the map t U(t)D.
The definitions and theorems in Dinculeanu [6] are given in
somewhat more generality; we restrict ourselves here to statements
involving the measures and ofields we shall be working with.
For two functions U1,U2: T L(E,F) we shall write U1 = U if
= pa.e. for every xEE, zeZ.
Let p be a lifting of L (u).
Definition 1.5.5. Let U: T + L(E,F) be a function.
a) We shall write p(U) = U if for every xEE and zEZ we have
E L (1) and
p() =
b) We shall write p[U] = U if there exists a family A of subsets
of T such that T has the direct sum property with respect to
A, such that for every AcA, xeE, zeZ, we have
1A E L () and
p(1 ) = p(A)
(Note: If p is ofinite (and in particular if p is a probability
measure), the relation in (b) holds for all A measurable.) The
functions U with p(U) = U or p[U] U have the following properties:
1) p(U) = U implies p[U] = U.
2) If p[U] = U, then is umeasurable for every xcE,
ZEZ.
3) If U U2, p(U ) = U, p(U2) = U2, then U1 U2'
4) If U1 U2, p[U ] = U1 and p[U2] = U2, then U1 = U2 pa.e.
5) If U = U2 pa.e. and p[U ] = U1, then p[U2] = U2'
We shall also make use of the following:
Proposition 1.5.6. Let U: T + L(E,F). If p[U] = U, then the
function t + IU(t) is pmeasurable. It is this proposition, along
with property (5) above, that will be used most often later.
1.5.3 RadonNikodym Theorems
We state here some generalizations of the RadonNikodym theorem
to vectorvalued measures with finite variation. These particular
statements are taken from Dinculeanu [6, pp. 263274]. Throughout
this section, T denotes a set, R a ring of subsets of T, E and F
Banach spaces, and Z a subspace of F', norming for F.
Theorem 1.5.7. Let m: R 4 L(E,F) be a measure with finite variation
p. If p has the direct sum property, then there exists a function
U : T 4 L(E,Z') having the following properties:
m
1) IU (t)I = 1 pa.e.
2) For all f E L (m) and zeZ, is pintegrable and we have
fdw.
3) If p is a lifting of L (u), we can choose U uniquely so
that p(U ) = Um (cf. Definition 1.5.5).
4) We can choose U (t) E L(E,F) for every t c T, in each of the
following cases:
a) F= Z'
b) There exists a family A covering T such that u has the
direct sum property with respect to A such that for
every A c A x e A, the convex equilibrated cover of the
set (fA ixdm: H Rstep function, JfAl dl 5 1) is
relatively compact in F for the topology o(F,Z).
c) For every x E E, the convex equilibrated cover of the set
{fixdm: l Rstep function, IfIidu S 11 is relatively
compact in F for the topology o(F,Z).
Note: If m is defined, on a oalgebra F, which will be the case in
our uses of this theorem, then the condition that T have the direct
sum property is automatically satisfied, and we can replace the family
by F in part (4b) of the statement.
These remarks also hold for the following generalization of the
RadonNikodym Theorem.
Theorem 1.5.8 (Extended RadonNikodym Theorem). Let v be a scalar
measure on R and m: R L(E,F) a measure with finite variation U. If
v has the direct sum property and if m is absolutely continuous with
respect to v, then there exists a function V : T + L(E,Z') having the
following properties:
1) The function IV m is locally* vintegrable and
fdy = IV mId ]vI for ip E L1 ()
(here uIv denotes the variation of v).
2) For f e Lc () and z E Z, is vintegrable, and we have
= fdv(t).
SAs before, If the measures are defined on an oalgebra, this can be
dropped.
3) If p is a lifting of L (u), we can choose V uniquely valmost
everywhere such that p[V ] = V (of. Definition 1.5.5). If,
m a
in addition, there exists a>0 such that p 'lvl, then we can
choose V uniquely such that p(V ) = V .
m a m
4) We can choose V (t) E L(E,F) for every t c T, in each of the
m
following cases:
a) F= Z'
b) There exists a family A covering T such that v has the
direct sum property with respect to A such that for every
A E A x c E, the convex equilibrated cover of the set
ifAJxdm: iRstep function, JAlI IdivI < 1I
relatively compact in F for the topology o(F,Z).
b') The same statement as (b), with A such that V has the
direct sum property with respect to A and with
IJidpj 5 1 instead of AI l@dlvl 5 1. In this case we may
not have p[V ] = V .
m m
c) For every x e E, the convex equilibrated cover of the set
iJfxdm: 4 Rstep function, IIIdlvl < 1} is relatively
compact in F for the topology o(F,Z).
c') The same condition as (c) with fIIJId 5 1 instead of
fl 4jdjvj 5 1. In this case we may not have p[V ] = V .
Theorem 1.5.7 gives a "weak density" of a vector measure m with
respect to its variation p, whereas Theorem 1.5.8, more generally,
gives such a density of m with respect to a scalar measure v not
obtained from m. The former is a particular case of the latter, but
we shall make good use of it in its own right, and so we take the time
to state it separately here.
The final result we shall need is a "converse" of these theorems.
Theorem 1.5.9. Let v be a scalar measure on R and U: T L(E,F)
a function such that JU' is locally vintegrable and the function
is vmeasurable for every x E E and z E Z.
Then the function is vintegrable for f E L1 (lUJDvi)
and z E Z and there exists a measure m: R L(E,Z') such that
= f<(t)f(t),z>dv for f E L1(Iu vl) and z E Z,
and Jljld J IUI ldlvl for L e Lll(uIV ).
The measure m has values in L(E,F) in each of the following cases:
a) F= Z'
b) For every x E E there exists a family A such that v has
direct sum property with respect to A such that for every
A E A the convex equilibrated cover of the set (U(t)x; t E A}
is relatively compact in F for the topology o(F,Z).
c) For every x E E, the convex equilibrated cover of the set
(U(t)x: t E T} Is relatively compact in F for the topology
o(F,Z).
d) The function t U(t)x is vmeasurable for each x E E; in
particular if F is separable.
If v has the direct sum property, we have the equality
S= IIIv!, hence
20
fJildu = flullldl vl for E L ()
in each of the following cases:
a) There exists a lifting p of L (v) such that plU] = U.
B) E is separable and there exists a countable norming subset
SC Z.
v) E is separable and the function t UI(t)x is vmeasurable for
every x E E.
6) The function U is vmeasurable (in this case we do not need
the direct sum property on v).
CHAPTER II
VECTORVALUED FUNCTIONS WITH FINITE VARIATION
Since the stochastic integral with respect to a process of finite
(or integrable) variation reduces to taking the Stieltjes integral
pathwise, it is appropriate to study functions defined on R2 (or R2)
with finite variation as a starting point. Throughout this chapter, f
will denote a function defined on R2 with values in a Banach space E,
unless explicitly stated otherwise. We shall write f(z) or f(s,t)
interchangeably for z = (s,t).
2.1 Basic Definitions and Some Examples
For functions of one variable, in order to associate an o
additive Stieltjes measure, we need the function to be either right or
left continuous. Here we shall use right continuity. In two
dimensions, however, there are two different notions of right
2
continuity: one for the order in R the other a condition merely
sufficient to ensure oadditivity of the associated measure.
Definition 2.1.1. Let f: R2 E be a function.
a) We say that f is right continuous (in the order sense) if, for
all z E R we have
f(z) = lim f(u), or equivalently lim If(u) f(z) = 0.
uz uz
u~z utz
(Note: We shall use sometimes the notation u + z for u z,
u 2 z.)
b) We say that f is incrementally right continuous if, for all
2
z R we have (denoting z' = (s',t'))
lim A z (f) = lim If(s',t')f(s',t)f(s,t')+f(s,t)
z'z (s',t')*(s,t)
z'"z s'"s
t'St
= 0.
Remarks.
1) The limits are pathindependent: in particular, in (a), this
limit includes the path where u z along a vertical or
horizontal path.
2) In (b) if s' = s or t' = t, A zz,(f) = 0, so we can take the
inequalities s' s, t' Z t to be strict. The chosen
definition is simply to preserve symmetry in the limits in (a)
and (b).
3) When we say simply, "f is right continuous," without further
specification, it will always mean in the sense of (a).
4) If f is right continuous, then f is incrementally right
continuous. To see this, note that IA zzfj = f(s',t')
f(s',t)f(s,t')+f(s,t) = If(s',t')f(s,t)+f(s,t)f(s',t)
f(s,t')+f(s,t)I (adding and subtracting f(s,t))
S lf(s',t')f(s,t) + If(s',t)f(s,t)j + Af(s,t')f(s,t) .
Then f right continuous implies each of the three terms on the
right tends to zero as (s',t') (s,t), hence A zzf 0,
which is (b).
Unfortunately, we do not have the converse implication in
general. To see this, we give the following example, which we shall
refer to later in pointing out further weaknesses of using increments
alone.
Example 2.1.2. Let g be any Evalued function defined on [0,=).
For (s,t) E R2, we then put f(s,t) = g(t). Then, for any
(s,t) $ (s',t'), we have
If(s',t')f(s',t)f(s(s,t )+f(s,t) = g(t')g(t)g(t')+g(t)A
= 0.
The function f is then evidently incrementally right continuous for
any f so defined. If we take, however, g to be a function which is
not right continuous, we have
lim jf(s',t')f(s,t)[ = lim jg(t')g(t)j 0,
(s',t')(s,t) t'4t
(s',t')a(s,t)
hence f is not right continuous. Later on, we shall establish some
additional conditions on f sufficient to have (b) => (a).
Another basic notion for oneparameter functions with regard to
Stieltjes measures is that of increasing function, as we reduce
functions of finite variation to this case via the Jordan
decomposition. Again, we have two definitions, the first the natural
extension of the onevariable definition (order sense), the second
more closely related to measure theory: namely, a condition
sufficient to generate a positive measure.
2
Definition 2.1.3. Let f: R R be a (realvalued) function.
a) We say f is increasing (in the order sense) if
z z' => f(z) S f(z').
b) We say f is incrementally increasing if A ,(f) 1 0 for
zz
all z z'.
The scalarvalued functions we shall typically consider are defined
using the variation of vectorvalued functions: these (as we shall
see later) are increasing in both senses. In general, however, the
two notions are distinctneither implies the other, as the following
two examples show. In these, we focus our attention on the unit
square [(0,0),(1,1)] for simplicity, but we can extend them (by
constants, say) to give a perfectly good counterexample defined on all
of R2
Examples 2.1.4.
i) We define here a function satisfying definition 2.1.3(a) but
not (b). The particular function we shall give is defined on the unit
square; we could extend it arbitrarily outside [(0,0),(1,1)], but we
shall not give an explicit extensionthe square is sufficient to
indicate how things can go wrong.
The idea consists of writing A ,f = f(s',t')f(s',t)f(s,t')+f(s,t)
as f(s',t')f(s',t)[f(s,t')f(s,t)], so if the second difference is
larger than the first, the increment will be negative even if f is
increasing in the sense of 2.1.3(a). Accordingly, for (s,t) in the
unit square, we define
f(s,t) = t + s(1 t).
Each onedimensional path, for fixed t, is a straight line connecting
the points (0,t,t) and (1,t,1). Thus, for t < t', the slope of the
section f(,t) is greater than that of f(,t'), and so the second
difference is larger than the first, so A ,f < 0 for any z < z' in
zz
the unit square. The following computations bear this out:
1) For (0,0) 5 (s,t) S (s',t') D (1,1), f(s',t')f(s,t) > 0. In
fact,
f(s',t')f(s,t) = t'+s'(1t')(t+s(1t))
= t's 's't'ts+st
= (t't)+(s's)+sts't'
= (t't)+(s's)+sts't+s'ts't'
= (t't)+(s's)t(s's)s'(t't)
= (1s')(t't)+(1t)(s's)
2 0,
since each of the four factors is nonnegative.
2) Denoting z (s,t), z' = (s',t'),(O0,0) (s,t) 5 (s',t')
S(1,1),
S,.f = f(s',t')f(s',t)f(s,t')+f(s,t)
zz
t'+s'(1t')(t+s'(1t))(t'+s(1t'))+t+s(1t)
= t'+s'(1t')ts'(1t)t's(1t')+t+s(1t)
= s'[(1t')(1t)]s[(1t')(1t)]
= (s's)(tt') 5 0.
ii) If we set g(s,t) = f(s,t), we get a function g satisfying
Definition 2.1.3(b), but not (a):
g(s',t')g(s,t) = f(s',t')(f(s,t))
= [f(s',t')f(s,t)]
5 0,
and similarly A ,g = (A ,f) > 0. We could even create a
zz zz
nonnegative g (g = 1f) with these properties.
As can be seen, these two definitions of increasing are not
nearly so closely related as the definitions of right continuity we
have given. Later, however, we shall give sufficient conditions for a
function f of two variables to have a "Jordan decomposition"
f ff 2, where f and f2 are increasing in both senses of the word.
2.2 The Variation of a Function of Two Variables
In this section we define the variation Var z,z',(f) of a
2 2
function f: R2 + E on a rectangle [z,z'] (closed) in R and establish
some of its properties. Throughout this section, by "rectangle" we
shall mean a closed, bounded rectangle in R2 (but everything goes
equally well for such rectangles in R ), unless otherwise specified.
Definition 2.2.1. Let R = [(s,t),(s',t')] be a closed, bounded
rectangle in R2
a) A partition P of R is a family of rectangles (R ) J
J JfJ'
finite, satisfying the following:
0 0
i) for j,j'cJ, j j', R. (h R = 0 (i.e., any two distinct
rectangles in (R.) are either disjoint or intersect only
J
on their boundaries)
ii) H = U R,.
JEJ J
(This is a straightforward extension of the notion of
partition of an interval [a,b]C R).
b) Let P = (R ) jc, Q = (R.) be two partitions of R. We say
j iJ l eT1
that Q is a refinement of P if, for each Rj E P, there exists
a family of rectangles from Q forming a partition of R..
J
(Note: It is evident from the definitions that for j j'.
The two families from Q forming partitions of R. and R., must
be disjoint.)
We show next that any two partitions of a given rectangle R have a
common refinement, as is the case in one dimension. The main step in
this, and a result we shall use again in its own right, is the
following:
2
Lemma 2.2.2. Let R [(s,t),(s',t')] be a rectangle in R2, and
P = (R )jEJ be a partition of R. Then there exist partitions
o: s = so < ss < ... < s = s' of [s,s'] and T: t = tO < tI <
... < t = t' of [t,t'] such that the family Q of rectangles of the
n
form [(s ,t ),(sp+1,t q)], O p < m, 0 5 q < n, is a refinement of P.
Remark. A partition of R constructed from partitions o,T of [s,s']
and [t,t'], respectively, as Q is above is called a grid on R. We
often use the notation oxx to denote the set Q of rectangles as
defined above as well as the vertices of these rectangles. There is
rarely any danger of confusion and where there is we shall be more
explicit. We shall use this notation in the proof of the lemma and
afterward.
Proof of Lemma. For each jeJ, denote R = [(s ,t ),(s',t)]. The
construction of Q is straightforward: we take o to be the set of all
the s. and s.,, ordered appropriately, and r to be the set of all
J J
t. and t.,, put in ascending order. We need to show that OxT is a
J J
refinement of P. Let, then, R. = [(s.,t.),(s',t')] be a rectangle in
P. Let a' be a partition of [s ,sj'] obtained by taking the points
from o between sj and s' (inclusive), and r' a partition of [t ,t']
obtained from T in the same manner. Then o'xT' is evidently a
partition of R.. I
J
Proposition 2.2.3. Let P,P' be two partitions of R. Then P and P'
have a common refinement, i.e., there exists a partition S of R that
is a refinement of both P and P'.
Proof. Let Q = oxx be a grid refining P (Lemma 2.2.2), and
Q' = o'xT' be a grid refining P'. Denote by p the partition of
[s,s'] formed by oo' (put in ascending order), Y the partition of
[t,t'] formed by putting Tr' in ascending order. Then S = pxY is a
common refinement of Q and Q', hence S refines P, and S refines P',
so S = pxY is our common refinement. (In fact, we have proved that
any two partitions have a common grid refinement.) I
For a given rectangle R, we can define an ordering on the class P
of partitions R as follows: we define P S Q for two partitions P,Q of
R if Q is a refinement of P. Prop. 2.2.3 says, then, that the class
PR of partitions of R is directed under this order.
We are now ready to define the variation of a function on a
rectangle.
Definition 2.2.4. Let R = [z,z'] = [(s,t),(s',t')] be a closed,
2 2
bounded rectangle in R and let f: R + E be a function.
a) For P = (Rj)) a partition of R, R = [(s ,t ),(s ,t )], we
~ JE J J i 3 3
define
Var zz'] (f;P) = IAR fl
JEJ J
E (f(sj,tj)f(s ,t )f(s ,tj)+f(s ,t ) .
jEJ
b) We define the variation of f on R, denoted Var z,z'(f), by
Var ,] (f) = sup Var (f;P) S +
[z,z ] [z,z']
PC
R
Remark. The supremum in part (b) always exists (finite or infinite),
since the map P Var[z,z (f;P) is increasing for the order defined
above on P To see this, consider a rectangle [z,z'] partitioned
into two rectangles R1 and R2, as in Figure 21.
t
t
s s5
Figure 21 A partition of [z,z']
We have
IA (f)l = f(s',t')f(s',t) f(s,t (s,t)f
= f(s',t')f(s',t)f(s ,t')+f( ,)+f()+f(s t')
f(sO,t)f(s,t')+f(s,t)
S f(f(s',t')f(s',t)f(so,t')+f(s0,t) + f(sg,t')f(so,t)
f(s,t')+f(s,t)
A fl + IA Rfl.
1R R2
We can do a similar calculation (only longer) for any partition of R,
by adding and subtracting values of f at all the additional vertices
of the refinement, and applying the triangle inequality.
In the next result, we give some properties of the variation.
2
Proposition 2.2.5. Let f: R + E be a function.
2
i) For any rectangle [z,z'] C R Var [z,z(f) 0.
ii) Var [z,z'(f) can be computed using grids, i.e., partitions of
the form Q = axT.
iii) Additivity: For 0 5 s < s' < s", 0 5 t < t' < t" we have
Var[(s, t),(st')] (f) = Var (s,t),(s )]
+ Var[(s',t),(st (f)
and similarly
Var[(st t f) = Var[(s,t),(s.t.)]
+ Var[(st (s',t")]
(See Figure 22.)
1 S Si
S s' st
Figure 22 Additivity of the variation
iv) If R1 C R2, Var (f) S VarR (f)
1 2
v) If f is right continuous (order sense), we can compute
Var [zz'(f) using grids consisting of points with rational
E zz 7
coordinates (and hence we can take the supremum along a
sequence of partitions).
Proof.
i) Since Var [z,'(f;P) Z 0 for any partition P, we have
Var [zz'](f) = sup Var z,z'](f;P) 0.
[zz'] [zz']
ii) If Var [zz](f) = + , then for every N>0, there is a
such that Varz,z By Lemma 2.2.2, there
partition PN such that Var .](f;P ) > N. By Lemma 2.2.2, there
N [z,z'N
exists a grid Q refining P ; by the remark following Definition 2.2.4,
we have Var ,(f;Q) a Var ,(f;P ) > N. Thus, if
[z,z ] [z,z ] N
Var[z,z ](f) = + , for any N>O there exists a grid QN such that
Var z, (f;Q N) > N, i.e., Var [zz] (f) = sup Var zz](f;Q).
[zz N [z.z ] Ez P .Z
'IR
Q=aox
Similarly, if Varz,z ](f) = a<, then for every E>O, there is a
partition P such that Var z (f,P ) > aE. Again, taking a grid
E [z,z'] E
Q refining P we have Var[ ,z(f;Q ) > Var z,'(f;P ) > ac.
C C [zz ] E [z,z'] C
e arbitrary => sup Var [ ,(f;Q) 2 a = Varz ,z(f). The other
QcP [z,z ] [z,z ']
QEPR
QoXT
inequality is evident, so we have Var[z,z'(f) = sup Varz,z(f;Q).
Q=oxt
(Note: From now on, we shall compute variations using grids.)
iii) We shall prove the first equality; the proof of the second is
completely analogous.
Denote R = [(s,t),(s',t')], R2 = [(s',t),(s",t')], R = R 1 R2.
For any grid Q = ox on R, we can add the point s' to o to get a
refinement Q' = Q U Q2, where Q is a grid on R Q2 is a grid
on R2. Then
VarR(f;Q) S VarR(f;Q') = VarR (f;Q1) + VarR (f;Q)
R R2
5 sup VarR (f;Q1) + sup VarR (f;Q2)
Q 1V Q( 2 2
= Var (f) + Var (f).
1 P2
Taking supremum on the left, we get
VarR(f) S Var (f) + VarR (f).
For the other inequality, if Q1 Q2 are any grids on R ,R2'
respectively, then Q U Q2 is a grid on R, and we have
VarR (f;Q) + VarR (f;Q) VarR(f;Q)
1 2
5 sup VarR(f;Q) = VarR (f).
Q=oXT
Since this inequality holds for any grid Q1 on R we have
sup VarR (f;Q1) + VarR2(f;Q2) S Var (f),
Q1 Ox 1 2
I .e.,
Var (f) + VarR(f;Q ) < VarR(f).
Similarly, Q2 being arbitrary, we have on taking supremum for Q2.
VarR (f) + VarR (f) 5 VarR(f).
Putting the two together, we have the equality:
Var (f) + Var (f) VarR(f).
1 2
iv) Assume R1 = [(s,t),(s',t')] is contained in
R2 = [(q,r),(q',r')]. Then we have q S s < s' s q', and
r S t < t' S r' (see Figure 23). By the additivity property, we have
VarR (f) = VarR (f) + VarR (f) + Var (f) + VarR (f) + Var (f).
2 1 3 14 5 6
Since each term on the right is nonnegative, we have
VarR2 (f) VarR (f).
B? B
v)
f(s,t) =
rl
q s s' q'
Figure 23 Decomposition of R2
Suppose, now, f is right continuous, i.e.,
lim f(s',t'). We show first that, for any grid Q = oxT
s'"s
t'+t
on R 00, there exists a grid Q' = o'xT' with rational coordinates*
such that
I 1 IRn fi E Z jfll < .
R EQ 1 R EQ' j <
Let, then, R [(s,t),(x,y)], a: s = s0 < s < ... < s = x be a
partition of [s,x], T: t = tO < t, < ... < tn = y be a partition of
[t,y]. We can choose points s5 > so, s; > a ,..., s > s and
t0 > to, t > t1... t' > tn so that, for each 0 5 k S m, O 5 1 < n,
we have If((sk t) ,t)f(s < k  We take then
o': s9 < < ... < and T': t < t' < ... < t'. Then for each
rectangle Rk [(s t ) (sk1 ,t +)] E Q, there corresponds a
* Note: Q' is not actually a partition of R, but here f is defined
outside of R so we can use these sums to compute the variation.
The main point of this is to show the variation is the limit of a
sequence, which we shall need later on.
rectangle R,1 = [(st'),(sCs + ,tl+1)] Q', and we have
IAR f IA fil A If AR f
k,l k,1 k, k,l
Sf(sk+l tl+ )f(sk+1 ,t)f(sk 1 )+f(s t)f(s+1 ,t +
+ (s + ,t;)+f(s',t + )f(s ,t')!
If(sk+1't l+1 )f(sk I't+l)l + If(Sk+1 tl)f( +l1't )
+ If(skt +1 )f(s ,t+1)1 + If(sktl)f(sktl)
C + + E:n+ C =CE:
4mn 4mn 4mn Nmn mn
Summing up, then, we obtain
E & fI E IAR fl = I  (AR f AR fI
R EQ 1 R .Q' O
OSi
Z Z IjA fj A fil < E E E.
k,l k,l k,l k,l mn
Now, if Var z,z',(f) = + *, for every N>O, there is a grid QN with
Var z,z](f;Q ) > N + 1. By the above, there exists a grid QN with
rational coordinates such that Var [zz'(f;QN) > N + > N,* hence
Lz,zJ N 2
sup Var ,z(f;Q) =+ Var[ ,z(f). Similarly, if
Q=oxT zz] zz
Q rational
Var[zz ](f) = a < m, for every e > 0 there exists Q = oxT such that
Var ,(f;Q ) > a and there exists Q' = o'xT' with rational
[z,z 'I E 2 E
* Again, Q' is not a partition of R, so this must be interpreted
directly as the sum given in Definition 2.2.4(a).
coordinates such that
IVar, '(fQ ) VarZZ](f;Q ) <
Lzz'] ar[z,z'] C t
hence
Var ,z (f;Q) > Var[zz(f;Q ) (a a .
[z,z] [zz E 2 2 2
c arbitrary => sup Var[ (f,Q) = Var ,](f). I
Q=z,z'] [z,z']
Q=oxT
Qrational
An important property of the variation in one dimension is that a
function of finite variation f is right continuous if and only if
lim Var (f) = 0 for all s in the domain of f. Unfortunately,
s 4s [s,s']
we do not have this equivalence in two dimensions without additional
assumptions about f. We do have one implication, however.
Theorem 2.2.6. Let f: R + E be a function with Var (f) < for
+ R
every bounded rectangle R. If f is right continuous, then for every
z, z', u in R2 with z < z' < u, we have
Var (f) = lim Var (f).
(Note: The notation u+,z' means u z', u > z'.)
Proof. We divide the region outside [z,z'3 and inside [z,u] into
three parts, labeled R1, R2, R (see Figure 24). The proof consists
r u
R3 R1
t
z
R2
t
zs s' p
Figure 24 Decomposition of [z,u]
of showing that, as u decreases to z', the variation on each of the
three rectangles R1, R2, and R3 vanishes. We shall give separate
proofs for R1 and R2, and the proof for R is identical to that
for R2. Note first of all, that for u' such that z' < u' < u, the
corresponding rectangles R', R2, R3 satisfy R1 C R R2 R2,
R C R3, so by Proposition 2.2.5(iv), VarRl(f) M Var (f), etc., and
1 1
so each of the limits lim Var (f), lim Var (f), lim Var (f)
u++z' 1 u++z' 2 u++z' 3
exists and is nonnegative.
a) Denote z' = (s',t') u = (p,r). We show that
lm Var[(s',t'),(p,r)](f) = 0.
r++t'
Assume not: then there exists a>O such that, for all u>z', we
have Var[zu](f) > a. Let u0 > z', E > 0. Denote u0 = (p0,r ).
Since Var[z',u > a, there exist partitions a : s' = ,0 < s, <
s0,2 < ... < sm = O' 'O: t' = t0,0 < tO,1 < .. < ton r0 such
that
i) z ( t )'Cs t fj > a
si
and OS
ii) ,,At) fjI < C by right continuity.
Ns( 0,0' 0,0),(s0,11't0,1) 2
(Recall that right continuity implies incremental right continuity.)
We have, then,
ZE IR fj + E IAR f + R fk > a 
RBOoXO xo0 B CO BR Oo 0
R CI R C II R C III
(See Figure 25.) Since each of the three sums is less than the
(S0,1,rO) (PO,ro)
I III
(s 't0, POl) l0,1
II
Si '(po,tI)
t,1
(s ,t')
s (sO, ,'t )
Figure 25 Computing variation of R1
variation of f on the respective rectangles, we have
+ V[ ]ar (f)
Var (s ,t ,1),'(s0,1 ,r0)](f) + Var[(s0,1',t'),(p t o,
+ Var (f) > c ,
P(s ,t1O' 0,1 0r0)]t)I 2
Denote u = (S0,1'taI) = (l ,rl) > z'. By assumption,
Var[z ,u (f) > a, so there exists a partition oa: s' = s1,0 < s1,1
< ... < s1,m p1 1: t' t,0 < t, < ... < t ,n = r1 such that
E JA f > a and 1r(s t )(s ,t f < Then, as
RBE:co 1 0,0't1,0 1,1
above, we have that the total variation of f on the three rectangles
comprising [(s' ,t'),pr) t'),(s ,t1,1 )] is greater than
a hence the total variation of f on the rectangles comprising
[(s',t'),(p0,rO)] \ [(s',t'),(s ,t, )] > (a ) + (a ) =
2, ( + ).
2 4
Denote u2 = (p2,r2) (s1,1't,1). By assumption,
Var[r ,u2 (f) > a, etc. Continuing in this manner, we construct a
sequence u0, u u2, ... with ui > ui+1 > z' for all i such that, for
all i, the total variation of f on the rectangles making up
[(s',t'),(pO,r0)] \ [(s',t'),(Pi,ri)] is greater than
i
(a ) + (a ) + ... + (a 21 ia ia E,
2 4 2 j=1 2
hence we have Var [z',u(f) > ia E for all i, i.e.,
Var z,u(f) = + , a contradiction of the hypotheses that
Var (f) < on every bounded rectangle. Hence, for any aO0, we have
0 lim Var, (f) < a => lim Var (f) = 0.
pts' uz' 1)(p)
r+4t'
This takes care of R1.
b) We show now that lim Var (f) = 0, or, more precisely, that
R
u+z' 2
lim Var[(s t(f) = 0. (See Figure 26). We proceed as
P +S [(s',t),(p,t')]
before, by contradiction. Assume there exists a>0 such that
(s,t') z/ (p,t')
R2
z(s,t) (s',t) (p,t)
Figure 26 Computing variation of R2
Var[(st)p,t')] (f) > a for all p > s'. Let, then, p0 > s', let
c>0. There exist partitions o0: s' = s0,0 < sO,1 < ... < Om PO'
TO: t = t0,0 < t,1 < ... < t,n = t' such that lA R f > .
RBEo~0xx0
Now, consider the rectangle [(s',t),(s ,1,t')]. For each i,
i 1,2,..., n, there is ri, s' < ri < s 0,1 ri+1 < r such that for
each i
[(s',toi1),(r ,t )] 1+1
by right continuity at each (s',t 1).
(rn,t') (so0,t') (Po,t')
t0,n1
0,2
I to,1
(s',t) rnr2 r1 (So01t) (Po,t)
Figure 27 Breakdown of 0oXT0
Subdividing each of the leftmost rectangles of a ox in this
manner creates a refinement P of o0XT0, hence E AR fi > a.
Now, we also have that
n n
i1 [( ,t0,i1),(rit0, f i=1 21+1 2 '
n
E A fI l E IA[, f > a ,
R P Y i=1 [(s ,t i1) ( ,t0l' ,i)] 2
The rectangles in this sum form a partition of the n rectangles
[(rl,t),(p,t0, )], [(r2',t 1),(P 0,2)], ... [(rn,tOn1),(p,t')]
(shaded rectangles in Figure 27). Thus
n
E Var (f)
i1 [( 't0,i1 0, l
n
> E A f E [(s, t ) )f I > 
R EP Y i=1 0,11 ) (rl 0,
In particular, then, Var > a Let p = By
[(r ,t),(pt')] 2 1 n
assumption, Var [',t),(r t f)n > a, so we can repeat this
procedure and get another point s' < r' < r such that
n n
Var[(r, t),( ,t.)](f) > a Continuing in this manner, we can
4
n 1
construct a sequence p0, p1' P2, such that
i) P > Pi+ > s, pi*s
ii) Var. t )(f) > a for all i.
L(p i+ t),(Pit )] 1+1
We have, then, by additivity,
i1
Var (f) = Z Var (f)
L(pi t),'(Pot )]f) J=O 1(pj+ t),(pj1t')
i1 i1
> E (a  ) = ia E  > a .
j=0 23 j=0 2 +1
11
Then Var (f) > E Var (f) > iaE.
2 j=0 Pj+1't)' t')J
E arbitrary => VarR2(f) > in for all i, hence Var R(f) = + , again a
contradiction. Hence lim Var (f) M a for any a>0, so
p++s" [(s' t),(pt')]
p++s
lir Varrst),p ](f) = 0.
By the same argument, lim Var(, ( (f) = 0 (the R
r t' s,t ,r)
case). Putting everything together, we have
lim Var C(f) = lim [Var ,(f) + Var (f) + Var (f) + Var (f)]
u++z' Lzu u++z' z ]1 R2 3
= lim Var ,(f) + lim Var (f) + lim Var (f) + lim Var (f)
u+z' [zz u+z, R1 u4+z' 2 u+4zt 3
= Var [z,z(f) + 0 + 0 + 0) = Var[z (f),
[z,z2] [2,25%
which is what was to be proved. I
Remarks.
1) As we stated above, the converse is not true. However, if
lim Var (f) = Var ,(f), then lim Var (f) = 0, and from
u+z [z,u [,z ] uz4,Z [z ,u]
IA[z',u]f f Var [',u](f) we get that lim [z. u]fl = 0, i.e., f is
u4+z
incrementally right continuous.
2) Returning to Example 2.1.2, for f as defined there, we have
Var[z,z'](f) = 0 on any rectangle [z,z'], since RE I~ (f)R =
R EP a
a 
Z 0 = 0 for any partition P of [z,z'], so lim Var (f) =
[z,uJ
R EP UZ UZ Lz
a 
Var[zz'](f). However, if f is constructed from a function g that is
not right continuous, then neither is f. In fact, for all e>0,
f(s+C,t+c) = g(t+E), so lim f(s+c,t+E) lim g(t+e) g(t) = f(s,t),
E+0 E40
so f is not right continuous. In the next section, we shall give
sufficient additional conditions on f for the converse to hold.
2.3 Functions of Two Variables with Finite Variation
As the example in the previous section (at the end) slows, the
requirement that VarR(f) < m on bounded rectangles R is by itself
insufficient to give all the properties necessary to associate a
Stieltjes measure to it. In order to deduce properties of f from its
variation, we need some extra conditions. It seems natural to require
that each of the onedimensional paths also have finite variation, but
we do not need quite that much. In fact, if VarR(f) < m on all bounded
2
rectangles RC:R and if the onedimensional path f(.,t ): R + E
has finite variation for some t0, then the paths f(',t) have finite
variation for all t. More precisely, for any s>0, we have
Var os]f(,t) < Var s]f(.,t ) + Var ( ,t ) ( (f).
(Note: We replace the second term by Var[(, t)(,t f) if
t < 0t). To see this, let o: O0 ( sO < s < ... < s s sbe a
partition of [O,s] (Figure 28): We have, for each i, 0 S i1 n1,
If(s i+ ,t)f(si,t)l = If(si+ ,t)f(s ,t)f(s +1 ,t0)+f(si,tO)
+f(si+ ,t0)f(sito)
SIf (s+ ,t)fst)(s ,t)f(si+,t)+f(si,to)
+ If(si+ ,t0)f(si ,t )
S1 S2  Sn1 s
Figure 28 Partition of bounding Var[0, f(,t)
Summing over the i's, and denoting Ri [(s ,t ),(s i1,t)], we have
n1 n1
S f(si+ ,tt), (si,t) E< ( 6Ri f + If(si+1,t0)f(si,t )0 )
i=1 i=1 i
n1 n1
= ~IA Rf + E If(s 1to)f(si ,tO)
i0 i 1=0
SVar[(O,t),(t)](f) + Var[ ]f(,t0)
(or S Var[(0,t),(St )(f) + Var[os]f(',t0) if t
Taking supremum over partitions a of [0,s], we get
Var[0,]f(.,t) < Var[(0,t ),(s,)](f) + Var o,s]f(,t ).
By the same proof (using partitions of [0,t]) we see that if the
onedimensional path f(s ,): R+ E has finite variation for some
SO, then the paths f(s,*) have finite variation for all f, and in fact
(same proof)
Var[ ,t]f(s,.) Var (s ),(s,) (f) + Var 0,t f(s, )0
Up to now, we have avoided using the phrase "f has finite
variation" because of the weakness of the condition Var (f) < m for R
bounded. We shall reserve this term for functions with the additional
conditions described above. We will see that this is enough to give
the additional properties we need to associate useful measures.
2
Definition 2.3.1. Let f: R2 + E be right continuous, with
2
VarR(f) < m for bounded rectangles R C R2. We say that f has finite
variation if the realvalued function
Ifj(s,t) = If(o,0)l+Var[os]f(*,0) arl o,t]f(o )+ar[( O),(s,t)]() <
2
for every (s,t) E R We say f has bounded variation if there exists
M>O such that
If (s,t) < M for all (s,t) E R.
The map If : R2+ R+ is called the variation of f. (Note that we use
the single bars to distinguish it from the norm in E.)
Remarks.
1) Henceforth, the phrase "f has finite variation" will be
2
understood to mean that IfI(s,t) < for all (s,t) E R.
2) We extend Ifl by 0 outside the first quadrant to get a
function defined on all of R2
function defined on all of R
3) The "jump at zero," If(0,0) will play a role later, similar
to that of the jump at zero In the theory of oneparameter
processes. When we associate measures with Ifl, we shall need this
term to get some compatibility between these measures and those
associated with f.
4) The function If is increasing in both senses of Definition
2.1.3. First of all, if z = (s,t), z' = (s',t'), and z < z', then
Ifl(s',t') Ifl(s,t) = 0 if (s',t') lies outside the first quadrant;
fl (s',t') fl (s,t) Ifl(s',t') > 0 if z' Z 0, z outside R2 If
0 S z < z', then
f(s',t') IfI(s,t) = If(0,0)I + Var[o, ]f(.,0)
+ Var[o,t']f(O,.) + Var[(o,0),(s',t')]f)
[f(o ,0)I + Var[os]f(,O) + Var [ ]f(0,.)
+ Var[(0,0),(s,t)] ]
= Var f(*,0) + Var [t,t f(0,.)
[ss ^ I[ttf 0
+ (Var[( ,),(s'.t')](f) Var[(o,),(s t)] )
0.
As for the other sense, we have A[z,z] fl = 0 if z' lies outside the
first quadrant. We then deal with the case where 0 5 z'.
If z is in the third quadrant, i.e., if s
= f (s',t') Ifl(s',t) fl (s,t') + Ifl(s,t) = Ifl(s',t') o 0.
If z is in the fourth quadrant, i.e., if s>0, t<0, then
A[zz.'](f) = If (s't') 0 If (s,t') + 0
= f(o,o)j + Var[o,s]f(,0) + Varo,t']f(O,)
SVar[(o0),(s',t')](f) (jf(0,0)j + Var s]f(.,0)
Var[ot']f(0,) + Var[(oo),(s,t )](f))
Var[s s ]f(,O) + Var[(s,0),(s',t')](f) 0.
Similarly, if z is in the second quadrant, i.e., if s<0, t>0, then
A[z,z'](f) = Var[t,t']f(O,*) + Var [(Ot),(s.,t')](f) o0. Lastly, if
0 S z < z', then we have
A[z,z' (f) f (s',t') f (s',t) fl(s,t') + If(s,t)
= If(0,0)j + Var[os']f(.,O) + Var[ot']f(O,)
SVar[(o,o),(s.,t.)] (f) (Ilf(O,0)I + Var1[os.]f(0)
Var[o,t]f(O,.) + Var[(o,o),(s',t)]
(If(0,0)j + Var1[os]f(.,0) + Varo,t']f(O,.)
SVaro[(0),(st')](f)) + f(,)I + Var[os]f(,)
+ Var[o,t]f(O,) + Var [(o,0),(st.f)]
(Va [(0,0),(s',t' )]( ) ar[(o, ),(s',t)](
(ar[(o,o),(s,t )]( ar[(o,0 ),(s,t)](
= Var (f) ( Van (f)
ar[(o.t),(s',t')] (ot),(st')]
= Var (f) 0.
= a [(s ,t ) ,(s '. t .)] f O .
This definition allows us to recover many results analogous to
those of functions of one variable, as we show in the next few
theorems.
Theorem 2.3.2. Let f: R2 E have finite variation Ifl. Then f is
right continuous if and only if Ifl is right continuous.
Proof. Assume, first, that f is right continuous. We write, for
(s,t) (0,0),
Ifl(s,t)
= If(0,0)l + Var o f(,O) + Vartf(O,) + Var [(O)(st)](f).
The first term is constant; to show that Ifl(s,t) = lim IfI(s',t'), it
s'+s
t'+t
suffices to show that
i) lim Var f(,O) = Var ,s]f(,O0)
s ++s ,s'] 10,s]
ii) lim Var [ ,t ]f(O,.) = Var ,t]f(0,.)
t' +t
iii) lim Var (O)( (f) = Var (f).
(s',t')++(s,t)]
We proved (iii) in Theorem 2.2.6 (taking z = (0,0), z' = (s,t),
u = (s',t')). As for the other two, f right continuous implies
f(',0), f(O,) right continuous: in fact, taking t' = t = 0, we have
lim
(s',t') (s,t)
(s',t')M(s,t)
f(s',t') = lim f(s',O) = f(s,0)
s'++s
(Recall that the definition of right continuity allows us to take
limits along vertical or horizontal paths as well, unlike left
limits.) Hence f(,O) is right continuous, so the variation is right
continuous, i.e., Var ,f(,O) = lim Var f(*,O). Similarly,
,stak = 0, we hav
taking s' = s = 0, we have
f(C,t) = f(s,t) 
lim
(s',t')(s,t)
(s',t')(s,t)
f(s',t') = llm f(0,t'),
t'++t
so f(0,) is right continuous. The variation is then right
continuous, so Var ,t]f(0,) = lim Var[,t ]f(0,). Then each of
t't ,t'
the terms of Ifl is right continuous; hence Ifl is right continuous on
R2
R.
Conversely, assume Ifl is right continuous at each point
2
(s,t) E R Taking t = 0, and letting s'4+s along the path t = 0, we
have
Var os' f(,O) Var[0,s]f(.,O) Ifl(s',0) Ifl(s,0).
In fact,
jfI(s',O) f(s,O) = f(O,O)I + Var o,s']f(,O) + Var ,f(O,)
+ Var( 0,), (s.O)]f (If(0,01 + var s]f(.,0)
Var[o,0]f(0,) + Var[(o,o),(s,0)f)
= Var[,sf(,O) Var[s]f(.,O).
Then Var [,]f(,0) = lim Var ,s']f(,0) since Ifl is right
[0,s] L0,s']
S'4 +S
continuous, i.e., the variation of f(,0) is right continuous; hence
f(*,0) itself is right continuous. A similar computation taking s = 0
shows that f(0,) is right continuous. Then, writing
Var[(0,0),(st)](f) If(s,t) jf(0,0) Var[os]f(.,0)
Varot]f(O,.),
we see that each term of If) is right continuous.
Now, let (s',t') (s,t), (s',t') (s,t). We have
If(s',t') f(s,t) If(s',t') f(s',t) + f(s',t) f(s,t)
S If(s',t') f(s",t)I + If(s",t) f(s,t)j.
We note now the following inequalities (cf. Figure 29):
1) If(s',t') f(s',t)j = If(s',t') f(s',t) f(0,t') + f(0,t)
+ f(0,t') f(0,t)I
S jf(s',t') f(s',t) f(0,t') + f(0,t)
+ If(O,t') f(0,t)
SIA[(o,t),(s',t')] fj + If(O,t') f(O,t)I
SVar[( ,t),( )](f) + If(Ot') f( ,t) .
2) If(s',t) f(s,t) = If(s',t) f(s,t) f(s',O) + f(s,O)
+ f(s',0) f(s,O)l
Sjf(s',t) f(s,t) f(s',O) + f(s,O)
+ f(s',O) f(s,0)I
SI[(s,),(s't)](f)l + If(s',0) f(s,3o)
SVar[(s,0),(s t)](f) + jf(s',0) f(s,0) .
Putting everything together, we have
If(s',t') f(s,t)l If(s',t') f(s',t)j + ~ f(s',t) f(s,t)
SVar[(ot),(s ,t )] (f) + If(Ot') f(Ot)I
+ Var[(sO),(s.'t)](f) + If(s'O) f(s0)
[Varf(O,),(s',t')](f) Var [( ,),( ,t)]
+ If(0,t') f(0,t)I + jf(s',0) f(s,0)D
(of. Figure 29). As we showed above, each of the three terms on the
Figure 29 Rectangles used in (1) and (2)
right tends to zero as (s',t') decreases to (s,t), so we have
lim If(s',t') f(s,t) = 0,
(s',t')+(s,t)
i.e., f is right continuous. I
Remark. We still have the same result if we extend f, Ifl by zero
2
outside R.
+
The next result concerns the existence of "onesided" limits and
"limits at infinity."
Theorem 2.3.3. a) Let f: R2 E be a function with finite variation
Ifl. Then each of the following limits exists* at each z = (s,t) e R2:
1) f(s@,t+) lim f(s',t')
t'tt
) f(s ,t ) = lim f(s',t')
s'+ts
t'+t
Sf(s+,t_) = lim f(s',t').
t'+tt
Moreover, if f has bounded variation (i.e., if there exists M>0 such
2
that Ifl(s,t) < M for all (s,t) E R ), then each of the following
* Of course, on the axes, not all these limits make sense. It will
be understood that at each point we take limits from quadrants
where f is defined.
"limits of infinity" exist:
1') f(s ,) = llm f(s',t')
s'+ns
2') f(s ,) = llm f(s',t')
t'+ts
3') f(=,t ) = lim f(s',t')
t'44t
4') f(m,t ) = lim f(s',t'), and especially
t '++t
5') f(m) = lim f(s',t') exists.
s',t't~
b) If, moreover, f is right continuous, then the onesided limits
along the vertical and horizontal paths f(s,), f(t) are equal to the
following.
i) lim f(,t) = lim f(s,') = f(s+,t+) (right limits)
s'4~s t'++t
ii) lim f(,t) = f(s,t+), and
s'tts
lim f(,t) = f(W,t+) if Ifl is bounded.
s 't
iii) lim f(s,) = f(s+,t), and
t'tt
lim f(s,) = f(s+,) if Ifl is bounded.
t'+
Remarks. 1) Here f is defined on R2: if we wish to use an f defined
also on the "boundary at infinity," the above limits will be denoted
with the symbol  in place of ".
2) In general, a function of finite variation can have eight
different limits at a point: the four "quadrantal" limits from part
(a) of the statement, plus the four onesided limits along the
vertical and horizontal paths. Part (b) of the statements says that
if f is right continuous, the onesided limits can be incorporated
into the quadrantal limits, so there are only four distinct limits at
a point (s,t) (at most). The first part of (b) says that both right
limits along the vertical and horizontal paths are equal to limit (1),
the second part says the left limit along the horizontal path is equal
to limit (2), and the third part says the remaining left limit is
equal to limit (4), giving the division of the plane shown in Figure
210. There are analogous considerations for the "limits at
infinity."
I
r
I
(s,t) (1) (2) (s,t)
(s t) (s t)
I
(3) (4)
Figure 210 The four "quadrants"
Proof. a) Assume first that f has finite variation If.
The proofs of limits (1)(4) are similar; we treat (1) first. We
shall show that for any sequence (s ,tn) with s *4s, t n+t, the
sequence (f(s ,t )}neN is Cauchy in E. We shall do this by denial:
n n neN
Let (sn,t ) be a sequence as above and assume that If(s ,tn)nN is
not Cauchywe shall reach a contradiction.
Since f(s ,t ) is not Cauchy, there exists e >0 and a subsequence
n n 0
(nk)kcN such that, for all k, we have
If(s ,t ) f(s t ) > E .
k+1 k+1 k nk
We will henceforth denote this sequence by If(sn,tn) n N, so we have
the inequality for all n.
For j given, consider the subdivisions I > s2 > ... > sj
> s = s of [s,s ] and t1 > t2 > ... > tj > tj, = t of [t,t ].
For i = 1,2,...,j1 denote Ri = [(s +,0),(sti+1)] and
R; = [(0,ti+1),(s ,t)] (Figure 211). For each i, we have
IAR fI = If(s i ) f(s i 1,t + ) f(si,0) + f(si+1,0), so
IAR fl If(si,ti+I) f(si+1t i+1)j f(s!,0) f(si+1,0) =>
If(si,O) f(si+1,0)I + IAf I f If(si,ti1) f(si+1,ti+1) We have
similarly
jAR fj If(siti) f(sti,+) f(O,ti) + f(O,ti+1)I
i
s If(si,ti) f(si,ti+1)I If(o,ti) f(0,ti+1
s j+si+'  s2
sj+1 s
Figure 211 Partition of [(0,0),(s,t)]
=> If((,t) f(O,t i+)+ Rf If(si,ti) f(si,t 1 ). Putting
the two together we have, for each i
!f(si,ti) f(si3 1,ti+1)l
= If(s ,ti) f(s ,ti+1) + f(siti+ ) f(s3i+l' t 1)l
< If(si,t ) f(si,ti+ )l + If(sl,t+ ) f(si+iti+1)
I AR. f R f + f f(si,0) f(si+ 10)I + Bf(O,t ) f(0,t)i+1
1 1
Now, upon summing over the i's, we have
j1
Z If(si ti) f(s i+1'ti+1 )
i=1
j1
E (JAR fT + f f (s ,O) f(si,+1,O)
i=1 i 1
+ If(0,ti) f(O,ti+1)
J1 j1
= (I A f + A Rfj) + E If(si,0) f(s +1,0)
i1 1 i i=1
j1
i=
S Var[(0,0),(s ,t (f) + Var[s s f(,O) + Var f[t t ')
(since the union of the R R; is contained in [(0,0),(sl,t1)])
SVar[(0,0),(st) (f) + Var [0s]f(*,O) + Var [,t]f(O,*)
S Ifl(sl,t ).
J1
We end up with Z f(s ,t ) f(si+1'ti+1)I fl(s ,tl). Now,
i=1
each term on the left is greater than EO, so we have
j1
Z If(s ,ti) f(si+1 ti+1) > (j1)EO, hence Ifl(sl,t1) > (j1)E0.
i=1
The left hand side does not depend on j, so letting j m, we obtain
SIfi(s ,t ) = + , a contradiction on the assumption that If < .
Thus, for any sequence (Sn,tn) decreasing to (s,t), the sequence
{f(s n,n ) N is Cauchy in E complete; hence lim f(s ,t ) exists. We
n n ncN n n
n
show now that we get the same limit for any sequence decreasing to
(s,t).
Consider two sequences (S ,t ) and (s',t'), both decreasing to
n n n n
(s,t). We construct a new sequence (p n,rn) as follows.
We set (P1,r ) = (sl,t ), (p2,r ) = the first term of (s3,t')
smaller than (p1,r ), (p ,r ) = the first term of (sn,t ) after
(s ,t ) smaller than (p2,r2), etc. The sequence (pn,r ) then
decreases and converges to (s,t) since both the even and odd
numbered terms do. Then L = lim f(p ,r ) exists from above. Looking
n
at the oddnumbered terms, we have L = lim f(p 2k+,r 2k1). But the
k
oddnumbered terms form a subsequence of (sn,tn), and we know
f(s ,t ) converges, so L = lim f(s ,t ) as well. Similarly, since
n n n n
n
the evennumbered terms form a subsequence of (s',tn), we obtain
n n
L = lim f(s',t') as well. The limit is then independent of the
n n
particular sequence, so lim f(s',t') exists, and it is
(s',t')++(s,t)
this limit we denote by f(s ,t ).
The proofs of (2)(4) are similar, and we will omit some
computational details where they are identical to the ones for (1).
Proof of (2). We consider a sequence (s ,t ) (s,t), with s n+s,
n n n
tn++t, and show that the sequence {f(s ,t )}nN is Cauchy in E, which
we again do by denial.
As in the proof of (1), we extract an E0 > 0, and (n ,tn) such
that (s ,tn) (s,t), and for each n we have s n > sn, tn+ < tn
and If(sn+1,tn+ ) f(sn,tn)I > 0E. As in (1), for given j, conside
the subdivisions s1 < s2 < ... < sj < s = s of [s ,s] and
t > t2 > ... t > t+ = t of [t,tl]. For i = 1,2,...,j1, denote
R = [(siO),(s ,t )], Ri = [(0,t i ),(si+ ti)]. (See Figure 212.)
We have, for each I (similar to before):
lAR fI f(si+1,ti) f(s ,t ) + f(s ,0)
> If(s ,t (s ,t ) If(s ,0) f(si,) ,
hence
tl
t2
t
t I
i+1 
R.
t (s,t)
Figure 212 Partition of [(0,0),(s,t)]
If(si ,0) f(s ,0) + JAR.fI If(si+ ,ti) f(si,t )I
1
= If(sit ) f(s i+ ,t t )
AR~fl = f(si+1,t ) f(s i+ ,t +1) f(0,t ) + f(O,ti+l
S f(si+ ',ti) f(si+1,ti+1) lf( ,ti) f(o,ti+ ) ,'
f(0,tC ) f(0,ti+l)1 + (IAR fj > If(si+ ,1ti) f(si+ ',ti+1) .
1
Here we diverge from the proof of (1), since the rectangles Ri, Ri
overlap (see Figure 212). From the first inequality we have, upon
summing over i,
j1
SIf(s ,ti) f(si+1 ,t )
i=1
j1
5 i ( R fI + If(si+1,0) f(si,0)p)
j1 j1
IE R f + E If(si+1,0) f(si'O)
i=1 1 i=1
S Var[(,),(s,1 (f) + Var s1s]f(.,0)
as in the proof of (1)
SI f (s,t )
(as in the proof of (1)).
Also, by a similar computation, we have
j1
Z If(si+lti) f(si+ ,ti+l)
i=1
j1 j1
5 I AR ,f + E If(0,ti) f(O,tl+)I
i=1 i i=1
Var[(0,0),(s,t l)](f) Var[t,t f(O )
SIfI(s,tl).
Putting the two together, we have (as in (1)):
j1
E If(silti) f(si+1'ti+1
i=1
j1
= If(siti) f(si+1,ti) + f(si+1,ti) f(s i+ ti+1)I
i=1
j1 j1
S If(si ,t ) f(s ,1 ti)I + E If(si+1,ti) f(s i+1t +c)I
i=1 i=1
5 IfI(s,t1) + Ifl(s,t )
21fl(s,t ).
Again, each term on the left hand side is greater than cO, so we get
j1
(j1)c0 < Z If(sit) f(s i+ ,ti+1)I 21fi(s,tl). Letting j + ,
i=1
we get IfI(s,t1) = + a contradiction as in (1); hence the sequence
If(sn,tn)} is Cauchy in E complete, so lim f (n ,t ) exists. The
n
remainder of the proof is exactly the same as that of (1).
The proofs of the last two are the same as those of the first
two: to prove (3), we use the same method as that of (1) to get
j1
E f(si,ti) f(si+1,ti+1) ) IfI(s,t) (instead of Ifl(sl,tl)),
i=1
and to prove (4) we use the same method as that of (2) to get
j1
E If(siti) f(si+1,ti+1) I Ifl(sl,t) (instead of If (s,t )).
i=1
This completes the first part of the theorem.
Assume, now, that f has bounded variation, i.e., there exists M
such that IfI(s,t) < M for all (s,t) E R. We will prove the
existence of the "limits of infinity" in pretty much the same manner
as the proofs of the other limits: the main difference occurs in
using M instead of a particular value of Ifl to obtain a
contradiction.
Proof of (1') Let (s ,t ) be a sequence with s ++s, s n < n
n n n n+1 n
Q
V = for all n, t t+. We proceed by denial as above. The proof of
n
this is much the same as that of (4) (and (2)), with a slight
difference: proceeding in the same fashion as in (2), we obtain
j1 J1 j1
Z If(s ,ti) f(si+t f(s+ ) f(si,O)l
i1 i=1 1 i=1
5 Jf (sl,t )
(see Figure 213). However, the right side now depends on j, so we
must further majorize it by M:
tj
ti+1
ti
tI
S S S  S i+ i  s
Figure 213 Partition for the "limit of infinity" f(s+,)
j1
Z If(s ,t ) f(si+1,t i) S f (st ) S M.
i1
Similarly,
j1
E If(s +1,t ) f(si+1 .ti+1) f (s1,tj) < M.
i=1
Now, as before, we have
j1
(j1)c < E f(sit ) f(si+1,ti+1 I
i=1
j1 j1
SEr If(sstt ) f(si+1,t ) + E Df(si+1 ) f(si,ti1
i=1 i=1
and we get a contradiction as before. Then f(s ,t ) is Cauchy; hence
llm f(sn,tn) exists, and we prove the limit is the same for any
n
sequence the same way as before.
This illustrates the difference between the proofs of (1)(4) and
those of (1')(5'): We do the same computation for the limits at
infinity, but the value of Ifl turns out to be at a point depending on
j, so we further majorize it by M. For (3') we have
j1
Z If(si,t ) f(si+1,ti) S If(tl ,s ) M
i=1
J1
SIf(si+1'ti f(s tl+1'ti+1)I Ifi(tl ,Sj ) 5 M,
i=1
so as before
j1
(j1)E0 < If(sl,t ) f(si+1ti+1) I 2M,
i=1
and we conclude as above. For (2'), (4'), and (5'), we follow the
same computation as in the proof of (1) and obtain
j1
(j1)E0 E If(sit ) f(si+1,ti+1) : f]j(sj,tj) S M
i=1
and conclude as in the proof of (1). This completes the proof of (a).
Proof of (b). Assume, now, that f is right continuous (order
sense!). We shall deal with (u) and (ui) first).
Ad (ii). Denote L = f(s,t+), let e > 0. There exists 6 > 0 such
that for all (s',t') with s't, ss' < 6, t't < 6, we have
f(s',t')LI < j Let, then, s' < s with ss' < 6: since f is right
continuous, there exists a point (s",t') with s' < s" < s, t' > t,
t't < 6, such that If(s',t) f(s",t') < But we also have
2
jf(s",t')Lj < hence ff(s',t)LI < gf(s',t) f(s",t') +
If(s",t')L < + E Thus, lim f(s',t) = L = f(s,t+).
25 2'tts
The proof of the limit at infinity is much the same, except that
instead of ss' < 6, there is N such that for all s' > N, the
conditions hold. We then take s' > N, s" > s, and the remainder is
the same.
Ad (iii). The proof of (iii) is the same as that of (ii), with the
roles of s and t being reversed: for E > 0 there exists 6 > 0 such
that for all s' > s, t' < t, etc. The remainder is the same.
Ad (i). This follows immediately from the definition of right
continuity: all three limits are equal to f(s,t). We should remark,
however, that it is the same to define right continuity using the open
quadrant: the limits along the horizontal and vertical paths are then
the same as the "quadrantal" limit. In fact, for E > 0, choose 6 > 0
so that for s' > s, t' > t, s's < 6, t't < 6, If(s,t)f(s',t')j < .
Then for any s' > s with s's > 6, there is a similar 
2
neighborhood" for the point (s',t). Pick any point in the
intersection of these neighborhoods," and apply the triangle
2
inequality as before. I
Our next result concerns the existence of a "Jordan
decomposition" for functions of two variables with finite variation:
in two variables, we have two distinct definitions of "increasing,"
but our decomposition satisfies both.
2
Proposition 2.3.4. Let f: R + R have finite variation (again, in
the sense of Definition 2.3.1). Then we can write f = f f2, where
f and f2 are increasing in both senses of Definition 2.1.3, namely
a) for (s,t) S (s',t') we have f (s,t) S f (s',t') and
f2(s,t) f2(s',t')
and
b) for (s,t) < (s',t'), A[(st),(s t )] ) Z O and
A (s,t),(s',t')](f 2) 0.
2
Proof. For (s,t) c R+, set f (s,t) = Ifl(s,t), f2(s,t) =
f (s,t) f(s,t) = Ifj(s,t) f(s,t). In remark (4) following the
definition of Ifj (Definition 2.1.3), we showed that jfl is increasing
in both senses. We then have only to deal with f2.
a) Let (s,t) S (s',t'). We have
f2(s',t') f2(s,t) Ifl(s',t') f(s',t') (Ifl(s,t) f(s,t))
(Ifj(s',t') If (s,t)) (f(s',t') f(s,t)).
We shall show f(s',t') f(s,t) Ifj(s',t') Ifl(s,t). Denote by
R1 the rectangle [(O,t),(s',t')], by R2 the rectangle [(s,O),(s',t)]
(Figure 214). We have
f(s',t') f(s,t)
I f(s',t') f(s,t)I
= If(s',t') f(s',t) + f(s',t) f(s,t)l
s f(s',t') f(s',t) + f(s',t) f(s,t)j
= RA f + f(0,t') f(0,t)l + AR f + f(s',0) f(s,0)
5 Ap fI + Ilf(0,t') f(o,t)l + A 2f + If(s',0) f(s,0)
5 VarR (f) + Var[t f(O,*) + VarR2(f) + Var[ss']f(.,0)
(Var (f.) Var (f))
S( ar[(o,o),(s',t')]( ) a [(o,o),(s,t)]
+ Var [ttf(0,.) + Varss.]f(.,0)
= fl(s',t') If (s,t)
(cf. Remark 4 following Definition 2.3.1).
~t (s', t)
R /
1 ,
t (s,t)
R2
s s'
Figure 211 Bounding the difference f(s',t') f(s,t)
Then Ifl(s',t') Ifl(s,t) > f(s',t') f(s,t), hence
f (s',t') f2(s,t) = (Ifl(s',t') Ifl(s,t)) (f(s',t') f(s,t)) Z 0,
i.e., f2(s,t) f2(s',t').
b) Let (s,t) < (s',t'): we have
A[(s,t),(s',t )](f2) (s',t') f2(s't) f2(s,t') + f2(s,t)
SIf (s',t') f(s',t') (Ifi(s',t) f(s',t))
(Ifl(s,t') f(s,t')) + Ifl(s,t) f(s,t)
S(If (s',t') IfI(s',t) If(s,t') + IfI(s,t))
(f(s',t') f(s',t) f(s,t') + f(s,t))
= (s,t),(s'.t')] (If ) [(st),(s ,t.)](f)
= Var'(s,t),(s',t')](f) A[(s,t),(s',t')](
0.
Thus f2 is increasing in both senses, and the
Remark. Both 2.3.3 and 2.3.4 hold with f and
outside the first quadrant.
proof is complete. I
Ifl extended by zero
CHAPTER III
STIELTJES MEASURES ON THE PLANE
In this chapter we extend the classical correspondence between
functions of finite variation on the real line and Stieltjes measures
on the real line to the case of functions and Stieltjes measures on
2
R.
3.1 Measures Associated With Functions
2
Given a function f: R2 E with finite variation on bounded
rectangles, right continuous (in the order sense!) on R we can
associate a unique measure with finite variation. The statement and
proof we give are due essentially to Radu [16]. The statement is a
little more general than we really need, but no further difficulties
are encountered by this; we also use rightlimits instead of Radus's
leftlimits, but this is just a matter of choice. The term "bounded
variation" in the statement refers to the variation of f on rectangles
as in Definition 2.2.4; as we have seen, this is weaker than the
requirement that Ifl be bounded.
Theorem 3.1.1 (Radu). If the function f: R + E is of bounded
variation on R2 and if the right limit f(s+,t+) (cf. Theorem .3.3 for
definition) exists at each point (s,t) of R then there exists a
2
Stieltjes measure m on R with values in E, uniquely determined,
with finite variation, and such that for all rectangles
R = ((s,t),(s',t')] we have
m(R) = f(s'+,t'+) f(s'+,t+) f(s+,t'+) + f(s+,t+)
Proof. We give the proof in several steps.
1) Let 6 be the family of rectangles of R of the form
(z,z'], z
measure theory; In fact, for any sets S,T with semirings of subsets
S,T, respectively, the family of sets of the form AxB with
A E S, B E T is a semiring of subsets of SxT. Here S = T = R, with
S,T the semirings of halfopen intervals.
We define a set function o: 6 E by
o(R) = AR(f ) = f(s'+,t'+) f(s'+,t+) f(s+,t'+) + f(s+,t+)
for R = ((s,t),(s',t')] with (s,t) < (s',t').
2) a is additive on 6. Let R R2 E 6, disjoint with
R 1U R2 E 6. Now, R1 R2 is also a rectangle iff they "match up" on
one side (see Figure 31).
t
R2
t'
R1
L 
t
s s'
Figure 31 Additivity on 6
Denote
R = (s,s'] x (t,t']
R2 = (s,s'] x (t',t"]
(Figure 31). (The proof is the same if R2 is of the form
(s',s"] x (t,t'].) We have
m(R1 R2) = f(s"+,t"+) f(s+,t"+) f(s'+,t+) + f(s+,t+)
= [f(s'+,t"+) f(s+,t"+) f(s'+,t"+) + f(s+,t'+)]
+ [f(s'+,t'+) f(s+,t'+) f(s'+,t+) + f(s+,t+)]
(we added and subtracted f(s'+,t'+) and f(s+,t'+))
= m(R2) + m(R ).
3) o has finite variation on 6. We prove this by contradiction;
suppose there exists a rectangle J E 6 such that Jol(J) = + w (oGl
denotes the variation of o). Then, for each a>O, there exists a
finite family (Jh), h = 1,2,...n of disjoint rectangles from 6,
JhC J for all h, such that
n n
E Jo(Jh ) > a, i.e., E ~6 (f+) > a.
h= h=1 n
Denote J = ((sh,th),(s th)] for all h. We may, of course, assume
n
that J = ) Jh (so that 'Jh e 6). Let > O. Since f has
n=1 h
rightlimits everywhere, there exists a number p > 0 common to all the
vertices of all the Jh such that
If(sh+,th+) f(sh Pth P)I
If(sh+,th+) f(sh p,t'+p)I < C
Wf(sh ,t+) f(sh+p,t^+p) < and
f(s+,'th+) f(sh+P,th+p) < i for all h.
n
If we denote Ih = [(s +p,th+p),(s'+p,t'+p)], then I = ) Ih is a
h h h h h h=1
h=1
closed, bounded rectangle in R2, and the family P = {Ih: h = 1...n}
forms a partition of I according to Definition 2.2.1. Also for each
h, we have
IJh (f+) A (f)
h h
= If(s+,t+) f(s+ 'th+) f(sh+t+) + f(s th)
(f(sh+p,th+p) f(sh+Pth+P) f(sh+pi,t+p) + f(sh+Pth+P))I
= lf(s ,'th+) f(sh+P,th+p)) (f(sh+,t +) f(s+Pth +))
(f(sh+,th+) f(sh+P,t +p)) + (f(h+,t h+) f(sh+Pth+P))h
+ If(sh t+) f(sh+Pt'+P)I + If(sh+'th+) f(sh Pth+P)
E E E
 u+ + "
In particular, we have
IAI (f)I > IA (f+)g E.
h h
Upon summing over h, we obtain
Var (f;P) = E A (f) > E 6 (f+) nc > a nc.
h h h
Now, denote J = ((p,r),(p',r')]: we can take p<1, and decreasing with
e, so for any c, we have Ih = I C [(p,r),(p'+1,r'+1)]. Denoting
this latter rectangle by K, we have K I for all I (in general, I
depends on e), so VarK(f) > Var (f) > a ne. e arbitrary =>
VarK(f) > a. Now, the collection Jh depends on a, but they all have
union equal to J, so we can repeat the above procedure for any a and
keep I C K. Thus, VarK(f) > a for any a => VarK(f) = + m, a
contradiction on our assumption of finite variation of f. Then a has
finite variation on 6.
4) o is inner regular on 6. We observe first that, from the
fact that the rightlimit of f exists at each point of R2, it follows
2 2
that for each z E R e > 0, there exists z' c R with z
for any u with z < u < z' we have If(u+) f(z+)] S e. In fact, since
f(z+) exists, there exists r>0 such that for any points u,v>z, with
luzi
point with Izz'l < n, z
If(u+)f(z') IS Likewise, letting v**z, we have
lf(v)f(z') < => If(z+) f(z')I 5 Then [f(u+) f(z+)I
S E
f(u+) f(z')j + If(z') f(z+), I + E = EF
Now, let J E 6, J = ((s,t),(s',t')] with s0.
There exists a point (p,r) E J such that for any point (h,k) with
(s,t) < (h,k) < (p,r) we have If(h+,k+) f(s+,t+)l < ,
If(s+,t'+) f(h+,t'+) < f(s'+,t+) f(s'+,k+) < as in
Figure 32. (We can do this for each by the above, and we use a
common n in choosing our (p,r).)
(s,t') (h,t') (s',t)
I I
II
I
I I (p,r)
S (s',k)
(h,k)
(s, t~ ~ ~ T s,t)
Figure 32 Approximation of a rectangle from within
Let, then, K = [(p,r),(s',t')] compact. Any rectangle J' from
6 such that K C J'C J must be of the form J' = ((h,k),(s',t')]
with (s,t) < (h,k) < (p,r) (see Figure 32). We have, then,
jo(J) o(J') IA(f+) J,,(f+)l
= f(s'+,t'+) f(s+,t'+) f(s'+,t+) + f(s+,t+)
 (f(s'+,t'+) f(h+,t'+) f(s'+,k+) + f(h+,k+)I
< f(s'+,t'+) f(s'+,t'+)I + If(h+,t'+) f(s+,t'+)
+ f(s'+,k+) f(s'+,t+)l + If(h+,k+) f(s+,t+)
C E E E
hence o is inner regular on 6. It follows, then, by Proposition 19
[6, p. 314], that JoI is also inner regular on 6.
5) Let r(6) be the class of subsets MC R2 for which M J E 6
for any J e 6. Since a is additive on 6, Jla is additive on t(6)
(standard result from measure theory), hence Jol is additive
on 6C (6). We shall now denote Jo by V (for clarity in what
follows).
Let o, V be the additive set functions obtained (uniquely) by
extending a and V to the ring C generated by 6. We show next that
n
o has finite variation on C. In fact, let A E C. Then A = ) A ,
1=1
Ai disjoint, Ai E 6. We have
n n
lo(A)l lo( A )I =  I o(Ai ) 5 E lo(A1 ) =
i i=1 i1
n n n n
= E o(Ai) ) I V(A ) = E V(Ai) = V( Ai) = V(A).
11 i=1 i=1 i=1
Then lo(A)I V(A) for all A c C, so lal S V (since Joi is the
smallest positive measure dominating lo(.) ), hence o has finite
variation.
6) Since c is inner regular on 6, o is inner regular on C = R(6)
[6, Corollary to Prop. 7, p. 308]. We now show o is regular on C.
This follows immediately from the following proposition [6, p. 306]:
Suppose that the ring C satisfies the following
condition: for every set A E C there exists a set A' C C
such that AC Int(A').
Then a measure m is regular on C if and only if m is
inner regular on C.
We need to show that C satisfies the condition. Let A E C,
n
then A = l A Ai E 6 disjoint. For each i, denote
1=1
Ai = ((sit ),(s',t')]. Then Bi ((si1,ti1),(s+1,t +
belongs to 6: clearly A C Int B for each i; hence
n n n
A = i A C J Int B C Int (J B.) = Int B, denoting
i=1 i=1 i=1
B = B B. E C. Take A' = B.
We have now an extension o of o to C satisfying:
1) o is additive on C
2) o is regular on C
3) o has finite variation on C.
Then, by a standard theorem of measure theory, o can be extended
uniquely to a Borel measure m of finite variation. This measure
clearly coincides with o on 6, so the theorem is proved. I
Remarks.
1) The theorem proved by Radu is for Rn: we have restricted
ourselves to R2 to enhance the clarity of the proof, but Rn presents
no additional difficulties (except with notation!).
2) The theorem holds in particular for the situation we use:
2
that where f is defined on R with If bounded, and f extended by
zero outside the first quadrant.
2
Suppose, now, that f is defined on R right continuous, with f]
finite, and extend f by zero outside the first quadrant. As an
exercise, we shall compute explicitly the measure of some sets in R2
using the limits developed in Theorem 2.3.3. More precisely, we will
compute the measure of points, intervals, and some rectangles in terms
of the "quandrantal" limits of f.
2
i) Let (s,t) E R Denoting by mf the measure associated
with f, we compute mf ((s,t))). We can write {(s,t)} ( An'
n=1
where A denotes the rectangle ((p ,q ),(p',q')] with
(pn,qn) < (s,t) < (pn,qn), pn ++s q n+', pn's, q'+t (see Figure
33). If we decompose A into four parts, labeled IIV in Figure 33,
we see that as (pn,q') + (s,t), parts I, II, and IV vanish. We may,
An
(p',q )
n n
I i
III (s,t)
nIV
I IV
(P ,q0)
Figure 33 Approximation of {(s,t)J
therefore, consider the upper corner of A to be (s,t) for each n, so
that we can take An = ((pn,qn),(s,t)] without loss of generality. We
have, then, by oadditivity of m ,
m f((s,t)}) = lim f (An) = lim A (f)
n n n
= lim (f(s,t) f(pn,t) f(s,qn) + f( n,q ))
n
= lim f(s,t) lim f(pn,t) lim f(s,qn) + lim f(p nq
n n n n
since the individual limits exist by Theorem 2.3.3
f(s,t) f(s ,t ) f(s ,t ) + f(s_,t_).
If we note that by right continuity we have f(s,t) = f(s+,t+), we see
that the measure of a point is analogous to the measure of a "half
open" rectangle, except we use the four limits to compute the measure
of a point.
ii) We next compute the measure mf of intervals of the forms:
s)} x (t,t'], {s) x [t,t'], {s) x [t,t'), {s} x (t,t'), and the
analogous "horizontal" intervals.
We begin with the closed interval I = {s} x [t,t']. We have
I = Rn, where R are rectangles of the form ((sn'tn),(s',t)]
n Rn n n n n n
n=1
with (s ,t ) ++ (s,t), (s',t') + (s,t'). As before, we can take
(s',t') = (s,t') for all n, so that R = ((s ,t ),(s,t')] with
n < s, tn < t, (s ,t ) tt (s,t) (see Figure 34).
Sn n
(snit/) (s,t')
Rn
(s,t)
I
L
(S, tn) s tn)
Figure 34 Approximation of an interval by rectangles
We have, then,
mf(I) = lim mf(R)
n
n
lim(f(s,t') f(ns ,t') f(s,t ) + f(s ,t ))
= f(s,t') lim f(sn,t') lim f(s,tn) + lim f(sn'tn)
n n n
= f(s,t') f(s_,tt) f(s+,t_) + f(s_,t_).
Similarly, we can represent the interval J = [s,s'] x {tJ
as J = Rn, where Rn = ((sn,tn),(s',t)], sn < s, tn < t,
n1
(Sn ,tn) ++ (s,t) (Figure 35).
(Sn't) (s,t) (s',t)
(sntn) (s',tn
Figure 35 Approximation of a horizontal interval
Again, we have
mf(J) = lim m (Rn
n
lim (f(s',t) f(s',t ) f(s ,t) + f(s ,t ))
n n n n
= f(s',t) f(ss,t_) f(s_,t+) + f(s_,t_).
With these in hand, we can compute the measure of the halfopen and
open intervals: We write I{s x (t,t'] = I \ C(s,t)}, so
mf(Is}x(t,t']) = mf(I) mf(I(s,t)})
f(s,t') f(s_,t;) f(s ,t_) + f(s_,t_) (f(s,t)
f(s+,t_) f(s_,t ) + f(s ,t_))
f(s,t') f(s_,t') f(s,t) + f(s_,t+).
Similarly, [s] x [t,t') = I \ {(s,t)}, so that
mf(is} x [t,t')) = mf(I) mf({(s,t')})
= f(s,t') f(s_,t') f(s+,t_) + f(s_,t_)
(f(s,t') f(s+,t) f(s_,t;) + f(s_,t'))
f(s3,t') f(s_,tl) f(s ,t_) + f(s_,t_).
As for the open interval, we have {s} x (t,t') =
{s} x [t,t') \ {(s,t)}, so
mf({s}x(t,t')) = mf (slx[t,t')) mf ((s,t)l)
= f(s ,t') f(s_,t') f(s+,t_) + f(s_,t_) (f(s,t)
f(s ,t_) f(s_,t ) + f(s_,t_))
= f(s ,t') f(s_,t') f(s,t) + f(s_,t+).
We use the same method for the intervals with t fixed. We give the
results:
mf([s,s')x{t}) = f(st,t+) f(s',t_) f(s_,t+) + f(s_,t_
m ((s,s']xltl) = f(s',t) f(s',t_) f(s,t) + f(s,t)
mf((s,s')xjt}) f(s',t ) f(s ,t_) f(s,t) + f(s ,t_).
iii) We now compute the measure of certain rectangles in R
If we allow the possibility of each side being open or closed, this
gives 16 different rectangles, and we do not give explicit
computations for them all. We shall go into detail for only a few,
and indicate the procedure for the remainder.
First, we shall give the measure of an open rectangle
R = (s,s')x(t,t). We write R = R where R ((s ,t ),(s',t)]
Sn n n n n
n
with (sn,tn) (s,t), (n ,tn) tt (s',t') (see Figure 36).
(s',t'O
!^)
^n
Figure 36 Approximation of open rectangle
We have, then,
m (R) = lim m (R
SI n
mr(R) = lim mf(Rn)
= lim (f(s',t') f(s,t ) f(s ,t') + f(s ,t ))
n n n n n n n n
n
= lim f(s',t') lim f(s',t ) lim f(s ,t') + lim f(s ,t )
n n n n n
= lim f(s',t') lm f(sim f (t im f(s ,t') + lim f(s ,t)
s'ts n n n n S n' n n3 ns n
n n n n
t'ttt t +t t't+t t +t
n n n n
= f(s ,t') f(s ,t+) f(s+,t') + f(s,t).
We next compute the measure mf of a closed rectangle
R [(s,t),(s',t')] = [s,s'] x [t,t']. We can write R =QRn, where
n
Rn = ((n'tn),(s',tn)] with (Sntn) +t (s,t), (s,tn) + (s',t')
(see Figure 37). As before, when n w, the rectangles labeled IIII
vanish, and so by oaddltivity of mf we can take (s',t') = (s',t')
h n n
without loss of generality.
(s t')
r n n
I
I (s,t)
(sn' tn
Figure 37 Approximation of closed rectangle
We have
mf(R) = lim mf(R)
n
= lim(f(s',t') f(s ,t') f(s',t ) + f(s ,t )
n n n n
n
= f(s',t') lim f(s ,t') lim f(s',t ) + lim f(s ,t )
n n n n
s "ts t +tt s + s
n n n
t +tt
n
= f(s',t') f(s_,t') f(s',t ) + f(s_,t ).
With these in hand, to obtain the measure of other rectangles it is
simply a matter of adding or subtracting the appropriate intervals
that comprise the sides of the rectangle. We illustrate this
procedure (as well as check our work!) by using R and some intervals
to compute m (((s,t),(s',t')]). (Note that the rectangle
((s,t),(s',t')] (s,s'] x (t,t'].) We have, denoting this rectangle
by A (Figure 38),
(s,t ) (s',t')
4
I
I A
I
(st) :s',t)
Figure 38 The rectangle (s,s']x(t,t']
A = {([s,s']x[t,t']) \ ((s]x[t,t'])} \ I(s,s']xIt)l.
(Note that the points (s,t'),(s',t) do not belong to A!) Then
m (A) = m ([s,s']x[t,t']) t,) m (s m ((s,s']x{t})
= f(s',t') f(s_,tD) f(s ,t_) + f(s_,t_) (f(s,t')
f(s_,t') f(s ,t_) + f(s_,t_)) (f(s',t) f(s;,t_)
f(s,t) + f(s,t_))
= f(s',t') f(s,t') f(s',t) + f(s,t),
which is how m (A) was originally defined. The measure of other
rectangles can be computed similarly using the parts already
explicitly given.
3.2 Functions Associated With Measures
In this section we consider the converse problem, namely, given
an Evalued measure m on R2 with finite variation, is it possible to
associate a function with finite variation such that m m in the
sense of Theorem 3.1.1? The following theorem provides a partial
answer to this question.
Theorem 3.2.1. Let m: B(R2) E be a measure with finite variation
Iml. There exists a function f: R2 + E with VarR(f) < on bounded
R
rectangles R such that m is the measure associated with f by Theorem
3.1.1, i.e., such that for all rectangles R = ((s,t),(s',t')] we have
m(R) = AR(f) = f(s',t') f(s',t) f(s,t') + f(s,t).
Proof. Define f: R2 + E by f(s,t) = m((C,(s,t)])* for
(s,t) R 2. We show first of all that AR(f) = m(R) for bounded
rectangles R. We have, denoting R = ((s,t),(s',t')]:
AR(f) = f(s',t') f(s',t) (f(s,t') f(s,t))
= m(( (s',t')]) m((',(s',t)]) [m((=,(s,t')])
m((m,(s,t)])]
= m((",(s',t')] \ (",(s',t)]) m((=,(s,t')] \ (C,(s,t)])
= m({(<,(s',t')] \ (=,(s',t)]} \ {(",(s,t')] \ (,(s,t)]})
= m(R).
We now show that f has finite variation on bounded rectangles,
i.e., that Var (f) < = for bounded rectangles R = [(s,t),(s',t')].
Assume note: suppose there exists R [(s,t),(s',t')] such that
VarR(f) = + =. Denote by R the halfopen rectangle ((s,t),(s',t')].
Let o: s = sO < s < ... < s s' be a partition of [s,s'],
T: t = t < tI < ... < tn t' be a partition of [t,t'], and let
P = oXT be the corresponding partition of R (of. Prop. 2.2.5).
* Note: (,(s,t)] = z e R2: z 6 (s,t)J.
Now, for any a>0, there exists such a partition P such that
VarR(f;P) > a, i.e., E A [(s ),(s)(f) > a.
051O m1 i'j i+1' j+1
OjSn1
However, the rectangles ((si,t ),(si+ ,tj+)], 0 S i S m1,
0 5 j n1 are disjoint, and their union is contained in R,
so we have
a < A[(s t ),(s ~ t ]
OSi m1 ) '(Si+1 'tj+ )J
OSj n1
z I'j ((s +,t ) 1' j+ (f)l
SIml (R).
Thus we have Iml(R) > a. a arbitrary > ImJ(R) = + , a contradiction
since m has finite variation. Hence VarR(f) < for R bounded, and
the theorem is proved. I
Remarks.
1) We have said nothing about uniqueness of f. In the case of
functions on the line, f is determined within a constant by m (i.e.,
any other associated function g is determined by adding or subtracting
a constant from f), but here this is not the case. In fact, as we
have seen before (Example 2.1.2) that many completely unrelated
functions can have zero as its associated measure.
2) The oadditivity of m implies that f is incrementally right
continuous, but as we have seen in Chapter II this is insufficient to
imply right continuity in the order sense without imposing finite
variation on the onedimensional paths f(s;) and f(,t).
2
We return now to the situation with f defined on R right
continuous, with finite variation fl, both extended by zero outside
the first quadrant. We have an important equality we shall make use
of in the next chapter, which is given in the following.
Theorem 3.2.2. Let m be the Evalued measure associated with f, and
let mlfi be the realvalued measure associated with Ifl. Then mf has
finite variation Imfl and we have the equality
ml Imflt
Proof. We showed in Theorem 3.1.1 that mf has finite variation. The
real thing to be proved here is the equality.
Let S be the semiring of rectangles of the form
R = ((s,t),(s',t')]. We shall show first that mlfI = Imfl on S.
We must consider various cases.
First of all, if (s',t') < (0,0), then mfl (R) = Imf (R) = 0.
We will assume, then, in what follows, that (s',t') lies in the first
quadrant. There are four cases, according to what quadrant (s,t) lies
in.
1) Assume (s,t) lies in the first quadrant. Let o: s =
s0 < s1 < ... < S = s' be a partition of [s,s'], T: t = t0 < t
< ... < t t' be a partition of [t,t']. Denote P = oXT the
corresponding partition of R:
P =R ijRij = [(st j),(s i+stj+)], 0 i S m1, 0 5 j 5 nl
Denote by R. the corresponding halfopen rectangles
1,J
((si,t ),(s i+1, )]. (We shall use this notation in the other cases
as well.) We have
ZI m (R ) I) A (f) l VarR(f) AR(If[)
ij i,J ij
(cf. Remark 4 following Defn. 2.3.1), and the right hand side is just
mlfi(R). The family (R ,) forms a disjoint cover of R with
Ri R, so taking supremum we obtain Iml(R) g mL (R). For the
ij IN .
other inequality, let E>0. There exists a grill oxi such that
I A R (f) > VarR(f) E m (R) .
1,j i,j
But the left hand side is equal to E Im (Rij ) and the Ri
i,jj
forms a decomposition of R, so we have
lmf(R)l a Z Jmf(R1'j) = E Z  (f)l > if ,(R) E,
iJ lj l mi
i.e., imf(R)I > m fl(R) E. Letting E + 0 (neither side now depends
on the corresponding grill), we obtain Imf(R)J a m fl(R); hence
Imf(R) = m f (R).
2) Suppose now (s,t) lies in the second quadrant, i.e., s
tZ0. For any grill oxT, we can refine o by including zero if it is
not already there, so that we may take oxT with zero included in o,
and compute variations with these grills (Figure 39). Denote by k
the index where sk = 0: For i I k 2, Imf(Ri,j) = 0; we also have
(s ',t')
(St t)
(S ,t 4 (O4ft4+++ (S ,t)
Figure 39 The grid axT
Imf(Rk1,j)I = lf(sk'.tj+ ) f(sktj) f(sk1'tj+1) + f(sk1,t )l
= f(o,tj+ ) f(O,tj) .
Putting everything together, we get
n1
E I (Ri,j) = r rf(o,t + ) f(0,t i) + Z IA (f)
i,j j=1 i k i ,J
j>O
S Var[t,t']f(O,) + Var[(o,t),(s 't )]
= Mv (R),
since m l(R) = Ifl(s',t') If (s',t) Ifl(s,t') + If(s,t)
 Ifl(s',t') Ifl(s',t) = [lf(0,0) + Var[0,s']f( ,O) + Var[ot.]f(O,*)
+ Var[(oO)(s','()]) [If(0,o) + ar[ 'f(,O) + Var[o,t]f(O'.)
+ Var[(o,0),(s',t) f)] [Var[(o,o),(s't )]() Var[(o,),(s',t))
+ [Var [0,]f(0,*) Var[0,t]f(0)] = Var[(0,t),(st)]
+ Var t,f(O,). Taking supremum over grills oxT on both sides, we
get (as before) Imf(R)I 5 mfI (R).
On the other hand, for any E>0, there exists a partition T' of
n1
[t,t'] such that E If(f((,t O,t)I > Var f(0,) and
J Q J+1 J Lt,t J 2
j0
a grill o0xr0 of [0,s']x[t,t'] such that ZIAR (f) >
1,j
Var [,t), .)(f) We choose a common refinement T of T
[(0,t),(s',t')] 2
and T., and extend 00 arbitrari:.y to get a partition o of [s,s'].
Then, for the grill oxT, we have
n1
E Imf(Rj .)I = ZE f(0,tj+,) f(0,t ) + E A (f) I (as before)
l,j j=0 a XT i,j
> Var[t, f(O, + Var[(,), (f) 
[t,tfl 2 [(OIt)I(s',2
= m fl(R) E.
Since the left hand side is bourded above by Imfl(R) for any grill,
we have Imfl(R) > mlfl(R) c. Letting e 0 again, we get
Im f(R) mi (R), hence equality.
The next case proceeds similarly.
3) (s,t) in the fourth quEdrant: sa0, t<0. This time, we
refine T by including zero if necessary, and compute variations along
such grills. Denoting tk = 0 a. before, we have (same computation as
before):
m1
Z jm (R ) E If(s+1 ,0) f(s ,0)j + Z AR (f)
l,j i=f0 IO i,j
j>k
SVar[s,s']f(0,) + Var[(s,O),(s',t')](
m f[ (R)
(same as before). Taking supremum we get imf(R) ml f(R). The
proof of the other inequality is the same as that for case (2).
4) Finally, assume (s,t) < (0,0). We proceed similar to the
above, but this time we add zero to both o and T, and use these
partitions in our figuring of variations (Figure 310). Denote
S= 0, t1 = 0. For i < k 2 or j < 1 2, we have Imf(Ri,)I = 0.
(s ',t
I
L__
(s,t)
Figure 310 The grid oxT
For I = k 1, j = 1 1, we have mf(R ij)I = If(0,0) for
'3
i = k 1, j 2 1, we have Imf(R ,) I = If(0,t ) f(0,t) I. For
j = 1 1, i 2 k we have m f(Ri ) = (f(si+1,0) f(si,0)(, all as
before. Putting everything together, we have
m1
SImf(R ,j) = If(0,0)I + EZ f(s +1,0) f(s ,0)
i,j i=k
n1
+ E f(O,t ) f(0,tj) + E Ri (f)
j=1 ikk i,j
jl1
Sf(0,0) + Var [,]f(.,0) + Var[,t]f(O,')
+ Var[(oo),(s', (f)](
= IfI(s',t') = m f (R).
Taking supremum again, we obtain Imf(R) 5S ma f(R). The proof the
other direction is similar to the ones before: for )>0, we choose a
common o,T so that
ZEf(si+1,0) f(si,0)I > Var 0,sf(.,0) 
a
EIf(O,tj+) f(O,t )I > Var tf(0,.) an
1'],t' 3 and
E 6R (f)l > Var (f) (
oT AR ,j [(0,0),(s',t')] 3
We extend and arbitrarily to partitions ',T of s,s,j
We extend a and arbitrarily to partitions oa,T' of [s,s'%
and [T,T'], respectively. The same computation as before gives
E jmf(Rj ) = ff(0,0)I + Zlf(si+1,0) f(si,0)D + ZBf(0,t )
i,j 0 T
f(o,t j) + Z AR (f)
oxT i,j
> jf(0,0)l + Var[0,s f(.,0) + Var[Ot']
+ Var (f)
[(0,0),(s',t')] 3
Ifl(s',t') E = m lf(R) E.
Hence Imf(R)I > mlfl(R) e. Letting e 0, we obtain
Im (R)l a m f(R); hence equality.
This takes care of all the possibilities, so we have Imfl = mlf
of S. Moreover, both are oadditive on S; the first since mf is by
Theorem 3.1.1, the second since Ijf is right continuous by Theorem
2.3.2. Since Imfl, mlf are equal and oadditive on S, they are equal
on a(S) B(R2), and the theorem is proved. x
CHAPTER IV
VECTORVALUED PROCESSES
WITH FINITE VARIATION
An important part of the general theory of processes in one
parameter is the correspondence between processes of finite variation
and measures on R+xn (see for example Dellacherie and Meyer [5, VI.
6489] and also Kussmaul [10]). This correspondence finds
applications in the notion of dual projections of processes, which are
used in the theory of potentials and in decomposition of
supermartingales (see for example Dellacherle and Meyer [5, nos. VI.
71113], also Rao [17] and Metivier [11]).
In the oneparameter case, the extension of the correspondence to
Banachvalued processes is done in Dellacherie and Meyer [5]. In two
parameters, the correspondence for realvalued processes is stated
(more or less) in Meyer [12]; we shall presently give a more directly
applicable (for our purposes) version, along with a proof, as the case
of finite variation on R2 is more delicate (as we have seen). In
fact, many times, in the literature results are given for increasing
processes, and then extended by defining a process of finite variation
as a difference of two increasing processes. The method we use here
is a little more constructive.
4.1 Definitions and Preliminaries
Throughout this chapter, (Q,F,P) will denote a complete
probability space, (F ) a filtration of subofields of F
z R2
satisfying the usual conditions. We also assume (F ) satisfies the
z
axiom (F4) of Cairoli and Walsh [2] (see section 1.2). Throughout
this chapter we shall denote by M the product ofield B(R2)xF. We now
state some definitions we will use in this chapter. (Some are
restatements from Chapter I, but we will give them again here for
completeness.)
Definition 4.1.1.
a) A (twoparameter) stochastic function is a function (not
necessarily Mmeasurable) X defined on R2x. Here, X will have values
in a Banach space, usually either in a Bspace E, or in the space
L(E,F) of continuous linear maps from E into another Banach space F.
We will consider X extended by zero outside the first quadrant, as we
did for functions defined on R2
+
b) A (twoparameter) stochastic process is a function
X: R xn E, measurable with respect to M = B(R2 )xF. A process X is
2
called adapted if X : Q I E is F measurable for each z c R2 (see
z z
Millet and Sucheston [13] and Chevalier [3] for related notions). We
generally use the term raw or brut to refer to a process that is not
necessarily adapted, i.e., such that X is Fmeasurable for each
22
z E +.
For fixed w e Q, the map X (w): R2 + E is called a path of the
process. Each path is a function defined on the first quadrant,
process. Each path is a function defined on the first quadrant, so we
shall use the results from the earlier chapters in studying these
processes. In particular, the variation of a process is defined in
terms of its paths. We have the following definitions.
Definition 4.1.2.
a) Let X be a raw process. We call X a process of finite
variation if, for each w, the path X (w): R2 R is a function of
finite variation in the sense of Definition 2.3.1. We define, for X a
process of finite variation, a realvalued process lXI, called the
variation of X by the following:
2
for w e 0, z = (s,t) ER ,
XIZ[(w) = IX.(w)l(s,t)
= IX(o,0)(w) + Var[0s] X.(w)I(,0) + Var[ot]X.(w) (0,.)
SVar[(0,0), (s,t)] *IX.(w) ).
b) If the random variable XIL lim Xi(s,t) 5 + (which
t+*
exists since jXI is increasing in the order sense) is Pintegrable, we
say X has integrable variation.
In this chapter, we shall concern ourselves with processes of
integrable variation. We will consider them extended by zero outside
2
R as we did for functions earlier.
Remark. In the book by Dellacherie and Meyer [5], processes of finite
variation are defined as differences of increasing processes. In two
parameters, it seems we might have a problem with this, as we have two

Full Text 
PAGE 1
TWOPARAMETER STOCHASTIC PROCESSES WITH FINITE VARIATION 3Y CHARLES LINDSEY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988
PAGE 2
TABLE OF CONTENTS Page ABSTRACT i i i CHAPTERS I INTRODUCTION 1 1 .1 Notation 3 1.2 Fi It rat ions 1 1.3 Stochastic Processes 8 1.4 Stopping Points and Lines 8 1.5 Some Measure Theory 12 II VECTORVALUED FUNCTIONS WITH FINITE VARIATION 21 2.1 Basic Definitions and Some Examples 21 2.2 The Variation of a Function of Two Variables 26 2.3 Functions of Two Variables With Finite Variation 43 III STIELTJES MEASURES ON THE PLANE 69 3.1 Measures Associated With Functions 69 3.2 Functions Associated With Measures 85 IV VECTORVALUED PROCESSES WITH FINITE VARIATION 95 4.1 Definitions and Preliminaries 96 4.2 Measures Associated With VectorValued Stochastic Functions 112 4.3 VectorValued Stochastic Functions Associated With Measures 131 4.4 On the Equality m = p , , 1M V CONCLUSI ON 1^9 BIBLIOGRAPHY 150 BIOGRAPHICAL SKETCH 154 ii
PAGE 3
Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TWOPARAMETER STOCHASTIC PROCESSES WITH FINITE VARIATION BY CHARLES LINDSEY April, 1988 Chair: Nicolae Dinculeanu Major Department: Mathematics Let E be a Banach space with norm Â», and f: R + E a function with finite variation. Properties of the variation are studied, and an associated increasing realvalued function f is defined. Sufficient conditions are given for f to have properties analogous to those of functions of one variable. A correspondence f ** \x between 2 such functions and Evalued Borel measures on R is established, and + the equality  u  = y . . is proved. Correspondences between Evalued twoparameter processes X with finite variation x and Evalued stochastic measures with finite variation are established. The case where X takes values in L(E,F) (F a Banach space) is studied, and it is shown that the associated measure u takes values in L(E,F"); some x sufficient conditions for y to be L(E,F)valued are given. Similar results for the converse problem are established, and some conditions sufficient for the equality y  = p. , are given. iii
PAGE 4
CHAPTER 1 INTRODUCTION Families of random variables indexed by directed sets have been objects of study for some time. The most common by far, though, has 2 2 been R , and especially R , the first quadrant of the plane, as this is considered the most "natural" extension of the usual indexing set R or R . One of the principal objects of study relating to such processes has been the stochastic integral of processes indexed by the plane. Stochastic integration with respect to twoparameter Brownian motion has been studied extensively; the focus of more recent study has been the more general problem of extending the oneparameter theory of stochastic integration with respect to a semima^tingale. In one parameter, this is done by writing a semimartingale X as X = M + A with M a locally square integrable local martingale and A a process of finite variation (see, for example, Dellacherie and Meyer [5]). The major problem in two parameters is with the notion of "local martingale": the theory of stopping is not sufficiently welldeveloped to permit a definition with all the necessary properties. Some preliminary work has been done, however, using the notion of an increasing path (see, for example, Fouque [9] and Walsh [18]). A definition of a stopped
PAGE 5
process has also been given for square integrable martingales in Meyer [12], but this too is rather limited. Another area that has seen renewed interest is the study of processes with values in a vector space, especially in a Banach space (see, for example, Neveu [1M]). In Dinculeanu [7] the correspondence given in Dellacherie and Meyer [5] between processes with finite variation and stochastic measures is extended to the case where the processes have values in a Banach space, and in Meyer [12] the correspondence is stated for twoparameter realvalued processes. In the oneparameter theory, this result finds its use in applications to projections, what in turn (at least in the scalar case) is relevant to the decomposition of supermartingales and to semimartingales . This correspondence X ** y is given by (1.1.1) u (*) = E(f* dX ) X ; z z for $ scala: Â— valued, bounded process. We shall extend this correspondence to the case where X has integrable variation, with values in a Banach space E. Since the inner integral on the right side of (1.1.1) is computed pathwise, we begin by studying the properties of Banachvalued 2 functions f: R Â» E with bounded variation. In Chapter II, we develop the relevant properties of such functions, and in Chapter III 2 we show that to each such f we can associate a measure p : B(R + ) Â» E with finite variation, and we prove the equality (1.12) p f  = M f 
PAGE 6
(i.e., the variation of the measure associated with f is equal to the measure associated with the variation of f). In Chapter IV we prove that a stochastic measure can be associated with a process of integrable variation, in the same manner as in Dinculeanu [7], and we consider the converse problem: that of associating a function with a given stochastic measure. Unfortunately, the equality (1.1.2) is not preserved in general for processes and stochastic measures, so we end up by establishing some sufficient conditions for the equality to hold, 1 .1 Notation 2 The index set is R ; we shall sometimes consider functions and 2 processes extended by zero to all of R . In the rare cases where we look at points outside the first quadrant, we shall say so 2 explicitly. We denote points in R by z, u, w, v and their coordinates by the letters r, s, t, p. For example, then, we write z = (s ,t) , u = (p ,r) , etc. There are two notations commonly used in the literature for the 2 order relation in R : we adopt here the notation of Meyer [12]. For two points z = (s,t), z' = (s',t'), we have zÂ£z' iff s^s', tÂ£t'; z
PAGE 7
denote [z,Â°Â°) = {u: z^u}, and by R the rectangle [0,z] (or (=Â°,z] if 2 we are discussing all of R ; these cases will be stated explicitly when they occur). We shall also have need to state the coordinates of rectangles explicitly, we write, for example, (z,z'] = ( (s ,t ) , (s' ,t ') ] ( = {(p,r): s
PAGE 8
12 11 convention F = F = F (which may or may not equal F = VF or ' '" 3 S V = VFp. Â°Â°~ t We then set F = F C\ F , and verify that this family satisfies st s t the usual conditions. 1 2 11 Each F contains the Pnegligible sets, since F , F v contain st st them for all s,t. 2) If (s,t) < (s'.t'), then F C F , ,. st s t 11 2 2 In fact, F C F , since sÂ£s', F t C F , since t^t , hence s s t t F . = F 1 PiF^C ?\,(~\?l, = F ... st s t s t s t 3) We have F = (] F , ,. ( Note : In two parameters, (s,t)<(s ,t ) our usual definition of right limit is for (s,t) Â£ (s'.f), (s,t) * (s',t'), as we shall see later. The statement we shall prove is somewhat stronger.) Proof. One containment is evident: since F ^C F , , for st s t each (s'.t') > (s,t) by (2), F C O F , r #Â» s t , v s t (s,t)<(s ,t ) s >s t'>t Then A z F , for all s'>s, hence A c (\ F , = F . s * v s s s >s Similarly, A c F , for all t'>t, hence A e f] F , = F . 1 t's t'>t Putting the two containments together gives the equality.
PAGE 9
We say that the condition of commutation is satisfied if the 1 2 conditional expectation operators E( Â•  F ) and E( *  F ) commute, 12 2 1 i.e., if E( Â• I F F ) = E( Â• F F ). The product is then equal to 1 s' t ' t ' s i ? 1 E( Â• I F HF ) = EC Â• I F ). (In fact, denote, for f e LÂ„(P), E a Banach 1 s t ' st t 12 2 1 space with the RNP, g = E(f  F F ) = E(fF F ). Then g is measurable 2 1 with respect to F and F , hence g is F measurable. Also, for t s st A E F st , we have J A gdP = J A E(f  F^ F 2 )dP = / A E(fF^)dP = J A fdP since A e F = F 1 C\ F 2 . Thus, g = E(f F ).) All of the main st s t st results of the theory of twoparameter processes require the condition of commutation. The second (and more frequently used) way of describing filtrations on R is due to Cairoli and Walsh [2]. There, we begin + with a family If , z e R 2 } of subofields of F satisfying the z + following axioms: (F1) If zz Meyer . ) 1 2 We now define F =F =VF V , F. = F = V F . In place of s s 00 . st t *t st the condition of commutation, we impose the axiom 1 ? (Fit ) For each z, F and F are conditionally independent given z z F , i.e., for Y, F measurable, integrable, Y_ F z 1 z d z measurable, integrable, we have
PAGE 10
(1.2.1) E(Y Y F ) = E(Y F )E(Y f ). I c. Z I Z c. Z This condition is equivalent [4, 11.45] to (1.2.2) E(Y 1 F 2 ) = E(Y 1 F z ) for every F measurable, integrable r.v. Y, . The condition (F4) can z Â— l be seen to be equivalent to the condition of commutation as follows: (F4) => commutation : Let X be Fmeasurable, integrable. Then E(XF 1 ) is F 1 measurable, integrable. By (F4), E( E(X I F 1 ) I F 2 ) = 1 z z ' z ' z 1 2 2 E(E(XF )F ) = E(XF ). Similarly, E(X F ) is F measurable, 1 z ' z ' z ' z z integrable; hence by (F4) E(E(X F 2 )  F 1 ) = E(E(xF 2 )F ) = E(XF ). Thus E(E(XF 1 )F 2 ) = E(XF ) = E( E(X I F 2 ) I F 1 ) , i.e., the operators 1 z ' z ' z ' z ' z 2 1 EC * I F ) and E( Â• I F ) commute. 1 z ' z Commutation => (F4) : We shall assume commutation and prove that (1.2.2) holds. Let Y, be F measurable, integrable. We have, from the 1 z commutation condition, E(YjF z ) = ECYjF^F 2 ) = E(Y.,  F 2 ) , which is just (1 .2.2) . Although the difference between the two constructions is slight (the main difference being on the "border at infinity," where we do not necessarily have right continuity of the filtrations (F ) and p (F ) in the CairoliWalsh model), the condition most often imposed on the filtration in the literature is stated as the (F4) condition, although the notation used is more often that of Meyer.
PAGE 11
1 .3 Stochastic Processes Throughout this section, (Q,F,P) is a complete probability space, (F ) a filtration satisfying axioms (F1)(F^) (in particular, one Z zcRj constructed by the method of Meyer), and E will denote a Banach space equipped with its Borel ofield, denoted B(E). The definitions In this section are taken from [12], [7]. Definition 1.3.1 . A stochastic process is a measurable (i.e., a 2 2 (B(R )xF, B(E) )measurable) function X: R^xQ + E. We usually denote X(z,w) by X (w), and the mappings w * X (w) by X . Remark . It will sometimes be convenient to extend the index set to 2 all of R by defining X =0 for z outside the first quadrant, and F = the degenerate ofield for those z. When we wish to consider a process in this light, we shall say so explicitly. A brut or raw process is a process X such that X is Fmeasurable 2 for all z e R . X is called adapted if X is F measurable for all + c z z 2 z e R . + 2 A process X is called progressive if, for every z z R , the map (z,w) X (w) from [0,z]xP, * E is (B([0,z])xF , B( E) )measurable. z z ( Note : This definition comes from Fouque [9]; in Meyer [12] progressi vity is defined using halfopen rectangles [0,z). The definition we give here is the one in common use today.) 1 . i) Stopping Points and Lines 1.11.1 Stopping Points The notion of stopping plays a fundamental role in the theory of oneparameter processes. The obvious extension of the definition of
PAGE 12
stopping time to twoparameter processes yields what is known as a stopping point . 2 Definition 1.^.1 . A stopping point is a mapping Z: fl * R such that for all v e R , the set {z < v} belongs to F . We often denote Z = (S,T). The components S,T are, it turns out, 1 2 stopping times with respect to (F ), (F ), respectively, but this by itself is insufficient to guarantee that (S,T) is a stopping point. We do, however, have the following characterization [12]: 1 2 Theorem 1.^.2 . Let S be an (F )stopping time, T an (F )stopping 2 time. Then Z = (S,T) is a stopping point if and only if S is F T ~ measurable, T F measurable. Although stopping points have found some application recently (see, for example Walsh [18] and Fouque [9]), they are rather inadequate for the purpose of developing a theory of localization for twoparameter processes. To begin with, due to the partial order in R , we are not even assured that JZ>v} e F ! Also, if U and V are + ' v stopping points, UV may not be. Moreover, in one dimension, the graph of a stopping time T divides R xft into two components, namely the stochastic intervals [[0,T)) and [[T,Â°Â°)), whereas the graph of a stopping point Z does no such thing. Furthermore, an important realization of a stopping time is as the debut of a progressive set, and we have no analogous result in the plane.
PAGE 13
10 1.4.2 Stopping Lines p Let A C R xQ be a random set (the usual definition: 1 . is a + A measurable process). The open envelope of A, denoted (A,*), is the random set whose section for each w e P. is given by (A,)(w) = \J (z,Â«). zeA(w) Some properties of (A,Â») (proofs can be found in Meyer [12]): 1) If A is progressive, (A, 00 ) is predictable. 2) The interior of a progressive set A is progressive, from which we obtain, by passing to complements, that the closure of A is progressive . We designate in particular by [A, 00 ) the closure of (A,*) (i.e., for each w, we define [A,Â°Â°)(w) = (A,Â«)(w)). [A,") is progressive if A is progressive, by the above properties; it is called the closed envelope of A. The random set D A = [A,Â«)\ (A,) A is called the debut of A : it is progressive if A is progressive. Definition 1.4.3 . a) A random set Z is called a stopping line if it is the debut of a progressive set, i.e., if there is a progressive set A such that Z = D = [A,Â»)\ (A,). A b) A stopping line Z is predictable if it belongs (as a set) to the predictable ofield P.
PAGE 14
11 Remarks . 1) The set A in the definition is not unique: for example, A and (A,Â°Â°) always have the same debut. 2) This definition has drawbacks: for example, D is not necessarily adherent to A. 3) For an alternate way of defining stopping lines (as a map from Â£2 into a certain set of curves on R ), see Nualart and Sanz [15]. H) Since (A,Â«) is predictable, to say that a stopping line Z = D is predictable amounts to saying that [A, 00 ) is A predictable, which is an alternate way of defining predictable stopping times in one parameter (cf. IH, IV. 69]). We introduce a partial order on the set of stopping lines by defining H < K if (H,) 3 (K,Â»). We also make the convention H < K if (H,) D [K,Â»). The set of stopping lines is then a lattice for Â£: HK is the debut of (H,Â») U (K,Â») , and HK is the debut of (H,Â») O(K,Â«0. We say that a stopping line H is the limit of an increasing (resp. decreasing) sequence (H ) of stopping lines if [H,Â») = [\[H ,Â») n (resp. if (H,Â») = I J(H ,Â»)). In the first case, if we have v y n n [H,Â») = 0(H ,Â°Â°), we say that the sequence (h ) announces (foretells) n
PAGE 15
12 H: if there exists a sequence (H ) announcing H, we say H is n announceable (foretellable) . In the oneparameter theory, this is equivalent to being predictable (and in fact many authors define predictability in this fashion). However, we only have one implication for stopping lines, namely, that every announceable stopping line is predictable. In fact, if H is announceable, then ex. (H ) such that [H,Â») = M(H .Â«). n n n Each (H ,*>) is predictable, hence the intersection is predictable. By remark (4), this implies that H is predictable. Unfortunately, the other implication does not hold. For a counterexample, see Bakry [1]. Finally, we note the existence of a predictable crosssection theorem for stopping lines. The proof is essentially the same as for the onedimensional case. We denote by tt the projection 2 of R xQ onto Q. Theorem l.H.M (predictable crosssection theorem). Let A be a predictable set, and let e>0. There exists a compact* predictable set K satisfying the following: 1) K C A and PJK = 0, A * 0} < e 2) DÂ„ is announceable and KCD . K ft 1 .5 Some Measure Theory In this section we collect some results from measure theory which we shall make frequent use of later. * That is, the section K(w) is compact for each w.
PAGE 16
13 5.1 Monotone Class Theorems There are several versions of monotone class theorems, both for families of sets and for functions. The two stated here are the variants we shall use later. The statements are taken from Dellacherie and Meyer [4]. Theorem 1.5.1 . Let C be a ring of subsets of a. Then the monotone class M generated by C is equal to the oring generated by C. If C is an algebra, then M = o(C). Theorem 1.5.2 . Let H be a vector space of bounded realvalued functions defined on Q, which contains the constants, is closed under uniform convergence, and has the following property: for every uniformly bounded increasing sequence (f ) of positive functions from H, the function f = lim f belongs n to H. Let C be a subset of H which is closed under multiplication. The space H then contains all bounded functions measurable with respect to the ofield o(C). The most frequent application of this theorem comes when we wish to prove that a certain property holds for all bounded Fmeasurable functions; it allows us to reduce to the case where f is the indicator of a set. We shall see numerous examples of this theorem at work. 1.5.2 Liftings Let (T,I,u) be a measure space, p>0. Material for this section comes from Dinculeanu [6, pp. 199216]. We shall need these properties in Chapter IV.
PAGE 17
Ill Definitions 1.5.3 . * Let u be a positive measure. A mapping p: LÂ°Â°(u) * LÂ°Â°(y) is said to be a lifting of L (u) if it satisfies the following six conditions: 1) p(f) = f ua.e. 2) f = g ua.e. implies p(f) = p(g) 3) p(otf + 8g) = otp(f) + Sp(g) for a,S e R 4) f>0 implies p(f)*0 5) f(z) = a implies p(f)(z) at 6) p(fg) = p(f)p(g). We say that LÂ°Â°(u) has the lifting property if there exists a lifting p of LÂ°Â°(u). The following theorem affirms that, for probability measures P in particular , LÂ°Â°(P) always has the lifting property. Theorem 1.5.4 . If it has the direct sum property, then L (p) has the lifting property. Let, now, E and F be Banach spaces, T a set, and Z a subs pace of F', norming for F, i.e., such that Â— 1 for every yeF. i y ^F = sup iff: r zeZ I Â»F' For every function U: T Â» L(E,F) ( continuous linear maps E * F) , x: T + E and z: T + Z, we denote by the map t Â•* and by U  the map t * U(t). * The definitions and theorems in Dinculeanu [6] are given in somewhat more generality; we restrict ourselves here to statements involving the measures and ofields we shall be working with.
PAGE 18
15 For two functions U ,U : T * L(E,F) we shall write U U 2 if = u~a.e. for every xeE, zeZ. Let p be a lifting of L (p). Definition 1.5.5 . Let U: T > L(E,F) be a function. a) We shall write p(U) = U if for every xeE and zeZ we have e LÂ°Â°(y) and p() = b) We shall write p[U] U if there exists a family A of subsets of T such that it has the direct sum property with respect to A, such that for every At A , xeE, zeZ, we have 1 A e LÂ°Â°(u) and p(1 A ) = 1 p(ft) . ( Note : If u is ofinite (and in particular if u is a probability measure), the "elation in (b) holds for all A measurable.) The functions U with p(U) = U or p[U] = U have the following properties: 1) p(U) = U implies p[U] = U. 2) If p[U] = U, then is ymeasurable for every xeE, zeZ. 3) If U Â» U , p(U ) = U, p(U 2 ) = U 2> then u i = Ug. H) If U U , pCU^ = U 1 and p[U 2 ] = Ug, then U 1 = U 2 pa.e. 5) If U = U ? ua.e. and p[U 1 ] = U 1 , then p[Ug] = Ug. We shall also make use of the following:
PAGE 19
16 Proposition 1.5.6 . Let U: T > L(E,F). If p[U] = U, then the function t * U(t) is ymeasurable. It is this proposition, along with property (5) above, that will be used most often later. 1.5.3 RadonNikodym Theorems We state here some generalizations of the RadonNikodym theorem to vectorvalued measures with finite variation. These particular statements are taken from Dinculeanu [6, pp. 26327^]. Throughout this section, T denotes a set, R a ring of subsets of T, E and F Banach spaces, and Z a subs pace of F', norming for F. Theorem 1.5.7 . Let m: R Â» L(E,F) be a measure with finite variation p. If \i has the direct sum property, then there exists a function U : T * L(E,Z') having the following properties: m 1) U (t) = 1 ya.e. * m 2) For all f e L^(m) and zeZ, is yintegrable and we have E m <[fdm,z> = [dy. J ' m 3) If p is a lifting of L (y), we can choose U uniquely so that p(U ) = U (cf. Definition 1.5.5). K m m k) We can choose U (t) e L(E,F) for every t e T, in each of the m following cases: a) F = Z' b) There exists a family A covering T such that y has the direct sum property with respect to A , such that for every A e A , x e A, the convex equilibrated cover of the
PAGE 20
17 set I J \Jjxdm: i) Rstep function, {j^dy < 1 } is relatively compact in F for the topology o(F,Z). c) For every x e E, the convex equilibrated cover of the set {/ijjxdm: i> Rstep function, Jiidu < 1 } is relatively compact in F for the topology o(F,Z). Note : If m is defined, on a oalgebra F, which will be the case in our uses of this theorem, then the condition that tt have the direct sum property is automatically satisfied, and we can replace the family by F in part (Mb) of the statement. These remarks also hold for the following generalization of the RadonNikodym Theorem. Theorem 1.5.8 (Extended RadonNikodym Theorem). Let v be a scala" measure on R and m: R * L(E,F) a measure with finite variation y. If \> has the direct sum property and if m is absolutely continuous with respect to v, then there exists a function V : T * L(E.Z') having the K m following properties: 1) The function V  is locally* vintegrable and m \m\i = fV Udlvl for i> e L (u) ; j Â« rn (here  \>  denotes the variation of v) . 2) For f z li(]i) and z e Z, is vintegrable, and we have :[fdm,z> = [dv(t), * As before, if the measures are defined on an oalgebra, this can be dropped .
PAGE 21
3) If p is a lifting of L (p), we can choose V uniquely valmost everywhere such that p[V ] = V (cf. Definition 1.5.5). Tf, m m In addition, there exists a>0 such that u i av, then we can choose V uniquely such that p(V ) = V . m y m m 4) We can choose V (t) e L(E,F) for every t e T, in each of the m following cases: a) F = Z' b) There exists a family A covering T such that v has the direct sum property with respect to A such that for every A e A , x e E, the convex equilibrated cover of the set {Jijttdm: 4jRstep function, L*dv Â£ l} relatively compact in F for the topology o(F,Z). b') The same statement as (b) , with A such that y has the direct sum property with respect to A and with ^dp <> 1 instead of J A *dv Â£ 1. In this case we may not have p[V ] = V . m m c) For every x e E, the convex equilibrated cover of the set Ijtjjxdm: \\> Rstep function, ^dv ^ 1 } is relatively compact in F for the topology o(F,Z). c') The same condition as (c) with Jipdy ^ 1 instead of J\Â»dv 5 1. In this case we may not have p[V ] = V . Theorem 1.5.7 gives a "weak density" of a vector measure m with respect to its variation y, whereas Theorem 1.5.8, more generally, gives such a density of m with respect to a scalar measure v not obtained from m. The former is a particular case of the latter, but
PAGE 22
19 we shall make good use of it in its own right, and so we take the time to state it separately here. The final result we shall need is a "converse" of these theorems. Theorem 1 . 5. 9 . Let v be a scalar measure on R and U: T * L(E,F) a function such that u J is locally vintegrable and the function is vmeasurable for every x e E and z g Z. Then the function is vintegrable for f e L 1 (uJv) and z g Z and there exists a measure m: R * L(E,Z') such that = /dv for f e L (uflv) and z e Z, and Ji/du Â£ J U 1 1 ^  d  v  for
PAGE 23
20 f*du = Juipdv for
PAGE 24
CHAPTER II VECTORVALUED FUNCTIONS WITH FINITE VARIATION Since the stochastic integral with respect to a process of finite (or integrable) variation reduces to taking the Stieltjes integral 2 2 pathwise, it is appropriate to study functions defined on R (or R ) with finite variation as a starting point. Throughout this chapter, f 2 will denote a function defined on R with values in a Banach space E, unless explicitly stated otherwise. We shall write f(z) or f(s,t) interchangeably for z = (s,t). 2. 1 Basic Definitions and Some Examples For functions of one variable, in order to associate an oadditive Stieltjes measure, we need the function to be either right or left continuous. Here we shall use right continuity. In two dimensions, however, there are two different notions of right 2 continuity: one for the order in R , the other a condition merely sufficient to ensure oadditivity of the associated measure. 2 Definition 2. 1 . 1 . Let f: R + > E be a function. a) We say that f is right continuous (in the order sense) if, for 2 all z e R , we have f(z) = lim f(u), or equivalently lim f(u) f(z)[ = 0. u*z uÂ»z u^z uSz 21
PAGE 25
22 ( Note : We shall use sometimes the notation u + z for u + z, u > z.) b) We say that f is incrementally right continuous if, for all 2 z e R , we have (denoting z' = (s'.t')) lim a ,(f) = lira f (s' ,t ')f (a' ,t)f (s ,t')+f (s ,t ) z'*z (s ' ,t ')Â»(s ,t ) z'Sz s'Â£s t'>t 0. Remarks. 1) The limits are pathindependent: in particular, in (a), this limit includes the path where u + z along a vertical or horizontal path. 2) In (b) if s' = s or t' = t, Ba ,(f)[ = 0, so we can take the 11 zz " inequalities s' > s, t' Â£ t to be strict. The chosen definition is simply to preserve symmetry in the limits in (a) and (b). 3) When we say simply, "f is right continuous," without further specification, it will always mean in the sense of (a). 1) If f is right continuous, then f is incrementally right continuous. To see this, note that a ,f = f(s',t')f (s',t)f(s,t')+f (s,t) = f (s'.t')f (s,t)+f (s,t)f(s',t)f (s,t')+f (s,t) I (adding and subtracting f(s,t)) < flf(s',t')f(s,t)l + f(s',t)f(s,t)U + f(s,t')f(s,t). Then f right continous implies each of the three terms on the
PAGE 26
23 right tends to zero as (s',f) Â•+ (s,t), hence & ,fj Â•> 0, which is (b) . Unfortunately, we do not have the converse implication in general. To see this, we give the following example, which we shall refer to later in pointing out further weaknesses of using increments alone. Example 2.1.2 . Let g be any Evalued function defined on [0,Â»). 2 For (s,t) z R , we then put f(s,t) = g(t). Then, for any (s ,t ) $ (s' , t ') , we have f(s',t')f(s',t)f(s,t')+f(s,t) = g(t')g(t)g(t')+g(t)f = 0. The function f is then evidently incrementally right continuous for any f so defined. If we take, however, g to be a function which is not right continuous, we have lim f(s',t')f(s,t)j = lira g(t')g(t)J * 0, (s'.t'Ws.t) t'+t (s',t')>(s,t) hence f is not right continuous. Later on, we shall establish some additional conditions on f sufficient to have (b) => (a). Another basic notion for oneparameter functions with regard to Stieltjes measures is that of increasing function, as we reduce functions of finite variation to this case via the Jordan decomposition. Again, we have two definitions, the first the natural extension of the onevariable definition (order sense), the second
PAGE 27
2iJ more closely related to measure theory: namely, a condition sufficient to generate a positive measure. o Definition 2.1.3. Let f: R > R be a (realvalued) function. a) We say f is increasing (in the order sense) if z < z' => f (z) < f(z'). b) We say f is incrementally increasing if A ,(f) i for all z Â£ z*. The scalarvalued functions we shall typically consider are defined using the variation of vectorvalued functions: these (as we shall see later) are increasing in both senses. In general, however, the two notions are distinct Â— neither implies the other, as the following two examples show. In these, we focus our attention on the unit square [( 0, 0) , ( 1 , 1 ) ] for simplicity, but we can extend them (by constants, say) to give a perfectly good counterexample defined on all of R 2 . + Examples 2.1.H . i) We define here a function satisfying definition 2.1.3(a) but not (b). The particular function we shall give is defined on the unit square; we could extend it arbitrarily outside [( 0, 0) , ( 1 , 1 ) ], but we shall not give an explicit extensionÂ—the square is sufficient to indicate how things can go wrong. The idea consists of writing A ,f = f (s ' ,t ' )f (s ' ,t )f (s ,t ' ) + f (s ,t ) zz as f (s',t ')f (s',t)[f (s,t ')f (s,t)], so if the second difference is larger than the first, the increment will be negative even if f is incresing in the sense of 2.1.3(a). Accordingly, for (s,t) in the
PAGE 28
25 unit square, we define f(s,t) t + 3(1 t) Each onedimensional path, for fixed t, is a straight line connecting the points (0,t,t) and (1,t,1). Thus, for t < t' f the slope of the section f(Â«,t) is greater than that of f(Â«,t'), and so the second difference is larger than the first, so A ,f < for any z < z' in zz the unit square. The following computations bear this out: 1) For (0,0) < (s,t) < (s'.f) < (1,1), f (s',t')f(s,t) i 0. In fact, f(s',t')f(s,t) = t'+s'(1t')(t+s(1t)) = t '+s's't 'ts+st = (t 't) + (3 's)+3t"S't' = (t 't) + (s 's)+sts't+s'ts't ' = (t't)+(s's)t(s's)s'(t't) (1s')(t't)+(1t)(s's) > 0, since each of the four factors is nonnegative. 2) Denoting z = (s,t), z' = (s ' ,t ') , (0,0) < (s,t) < (s'.f) $ (1,1). ,f = f(s',t')f(s',t)f(s,t')+f(s,t) =t'+s'(1t')(t+s'(1t))(t'+s(1t'))+t+s(1t) = t'+s'(1t')ts'(1t)t's(1t')+t+s(1t)
PAGE 29
26 = s'[(1t')(1t)]s[(1t')(1t)] = (s's)Ctt') < 0. ii) If we set g(s,t) = f(s,t), we get a function g satisfying Definition 2.1.3(b), but not (a): g(s',t')g(s,t) = f(s',t')(f(s,t)) = [f(s',t')f(s,t)] $ 0, and similarly A ,g = (A ,f) Â£ 0. We could even create a zz zz nonnegative g (g = 1f) with these properties. As can be seen, these two definitions of Increasing are not nearly so closely related as the definitions of right continuity we have given. Later, however, we shall give sufficient conditions for a function f of two variables to have a "Jordan decomposition" f = f t , where f and fÂ„ are increasing in both senses of the word. 2.2 The Variation of a Function of Two Variables In this section we define the variation Var r ,,(f) of a [z,z J 2 2 function f: R Â•* E on a rectangle [z.z'J (closed) in R + and establish some of its properties. Throughout this section, by "rectangle" we 2 shall mean a closed, bounded rectangle in R (but everything goes 2 equally well for such rectangles in R ), unless otherwise specified. Definition 2.2.1 . Let R = [ (s ,t ) , (s ' ,t ' ) ] be a closed, bounded 2 rectangle in R .
PAGE 30
27 a) A partition P of R is a family of rectangles (R . ) . , J J J c J finite, satisfying the following: i) for j.j'eJ, j * j', R . C\ R . = (i.e., any two distinct rectangles in (R . ) are either disjoint or intersect only on their boundaries) ii) R = \J R . (This is a straightforward extension of the notion of partition of an interval [a,b](ZR). b) Let P = (R ) ,, Q = (R ) , be two partitions of R. We say J jeJ i iel that Q is a refinement of P if, for each R. c P, there exists J a family of rectangles from Q forming a partition of R . . ( Note : It is evident from the definitions that for j * j'. The two families from Q forming partitions of R. and R., must J J be disjoint.) We show next that any two partitions of a given rectangle R have a common refinement, as is the case in one dimension. The main step in this, and a result we shall use again in its own right, is the following: 2 Lemma 2.2.2 . Let R Â» [ (s,t) , (s ,t ) J be a rectangle in R , and P = (R.). T be a partition of R. Then there exist partitions J JeJ o: s = s^ < s, < ... < s = s' of [s,s'] and t: t * t n < t, < Dim u i ... < t = t' of Ct.t'] such that the family Q of rectangles of the n form [(s ,t ),(s . ,t . ) ], Â£ p < m, % q < n. Is a refinement of P. p q p+1 q+1 Remark . A partition of R constructed from partitions o,t of [s,s'] and [t,t']> respectively, as Q is above is called a grid on R. We
PAGE 31
28 often use the notation oxx to denote the set Q of rectangles as defined above as well as the vertices of these rectangles. There is rarely any danger of confusion and where there is we shall be more explicit. We shall use this notation in the p^oof of the lemma and afterward. Proof of Lemma. For each jeJ, denote R, = [ (s . ,t . ) , (s ' ,t ') ]. The J J J J J construction of Q is straightforward: we take o to be the set of all the s. and s ,, ordered appropriately, and t to be the set of all t. and t.,. put in ascending order. We need to show that oxt is a J J refinement of P_. Let, then, R. = [ (s . ,t . ) , (s ' ,t ' ) ] be a rectangle in _P. Let o ' be a partition of [s.,s '] obtained by taking the points from o between s. and s' (inclusive), and t ' a partition of [t.,t'] J J J J obtained from t in the same manner. Then o'xt' is evidently a partition of R , B Proposition 2.2.3 . Let P,P' be two partitions of R. Then P and P' have a common refinement, i.e., there exists a partition S of R that is a refinement of both _P and P'. Proof . Let Q = oxx be a grid refining _P (Lemma 2.2.2), and Q' = o'xt' be a grid refining P". Denote by p the partition of [s,s'] formed by oo' (put in ascending order), Y the partition of [t,t'] formed by putting tt ' in ascending order. Then S = pxY is a common refinement of Q and Q', hence S refines _P, and S refines P', so S = pxY is our common refinement. (In fact, we have proved that any two partitions have a common grid refinement.) B For a given rectangle R, we can define an ordering on the class Pp of partitions R as follows: we define P S Q for two partitions P,Q of
PAGE 32
29 R if Q is a refinement of P_. Prop. 2.2.3 says, then, that the class P D of partitions of R is directed under this order. We are now ready to define the variation of a function on a rectangle . Definition 2.2.4 . Let R = [z,z'] = [ (s ,t) , (s' ,t ') ] be a closed, 2 2 bounded rectangle in R , and let f: R Â» E be a function. a) For P = (R.) a partition of R, R = [ (s . ,t . ) , (s ' ,t ' ) ] , we J JeJ J J J J J define Var rz z'i (f; ^ = l K f ' Lz.z j jeJ Kj = X f(s' t')f(s' t )f(s ,tp + f(s ,t )J. 4_T JJ JJ JJ JJ b) We define the variation of f on R, denoted Var r ,,(f), by Lz,z J Var r ,,(f) = sup Var r ..(f;P) S + Â». Lz,z J [z,z ] ? C R Remark . The supremum in part (b) always exists (finite or infinite), since the map P > Var^ ,, (f;Â£) is increasing for the order defined above on P . To see this, consider a rectangle [z,z'] partitioned R into two rectangles R and R , as in Figure 21, R 2 I R l Figure 21 A partition of [z,z']
PAGE 33
30 We ha ve A (f) = f(s',t')f(8',t)f(s,t')+f(3,t) = f(s',t')f(s',t)f(s ,t')+f(s ,t)+f(3 ,t') f(s ,t)f(s,t')+f(s,t)l < Jf(s',t')f(s',t)f(S ,t')+f(S ,t)l + f(8 ,t')f(B ,t) f(s,t')+f(s,t) = A R f + A R f. We can do a similar calculation (only longer) for any partition of R, by adding and subtracting values of f at all the additional vertices of the refinement, and applying the triangle inequality. In the next result, we give some properties of the variation. 2 Proposition 2.2.5. Let f: R + E be a function. E + 2 i) For any rectangle [z,z ] C R , Var,,,(f ) Â£ 0. ii) Var r ,,(f) can be computed using grids, i.e., partitions of the form Q = oxt. iii) Additivity : For < s < s' < s" , Â£ t < t' < t" we have [ (s,t) ,(s",t ) ] L(s,t) ,(s ,t ) J + Var r . , t , , Â„ ,. ,(f ) [(s ,t) ,(s",t ) ] and similarly
PAGE 34
Var [(s,t)(3',t")3 (f) = Var [(s,t),(3',t')] (f) +Var r( S( t') )( s' ( t")] (f) (See "igure 22.) 3 1 t".t 't Figure 22 Additivity of the variation iv) If R,C R_, Var D (f) < Var D (f). 1 2 R 1 R 2 v) If f is right continuous (order sense) , we can compute Var> ,,(f) using grids consisting of points with rational Lz,z J coordinates (and hence we can take the suprernum along a sequence of partitions). Proof . i) Since Var r ,,(f;P) Â£ for any partition P, we have [z,z ] far r Â„(f) = sup Var r ,,(f;P) ^ 0. [z,z ] D [z,z J ii) If Var r , n (f) = + Â», then for every N>0, there is a Lz,z J partition P such that Var (f;P ) > N. B Y Lemma 2.2.2, there N [z,z'] Â— N
PAGE 35
32 exists a grid Q refining P Â• by the remark following Definition 2.2.4, we have Var r , n (f;Q) Â£ Var r , n (f;PÂ„) > N. Thus, if L z , z J L z , z J Â— N Var r ,n(f) = + Â°Â°, for any N>0 there exists a grid Q such that L z , z ] N Var Cz,z'] (f; V > N ' Ue " Var [z,z'] (f) = *Â£ p Var rz,z'] (f;Q) ' Qer R Q=OXT Similarly, if Var r ,,(f) = ctO, then for every e>0, there is a [z,z ] partition P such that Var r ,,(f,P ) > ae. Again, taking a grid e [z,z J 'Â— e Q refining P , we have Var r ,,(f;Q ) > Var r ,,(f;P ) > ae. e e [z,z J e [z,z J e e arbitrary => sup Var r , n (f;Q) S a = Var r , n (f). The other p Lz.zJ Lz.zj Q=OXT inequality is evident, so we have Var r ,,(f) = sup Var r ,.(f;Q) Lz.z j Qe p R Lz.z J Q=OXT ( Note : From now on, we shall compute variations using grids.) iii) We shall prove the first equality; the proof of the second is completely analogous. Denote R = [ (s ,t ) , (s ' ,t ' ) ] , R 2 = [ (s ' ,t ) , (s" ,t ' ) ], R = R \J R^ For any grid Q = oxt on R, we can add the point s' to a to get a refinement Q' = Q \J Q , where Q is a grid on R , Q is a grid on R . Then Var R (f;Q) < Var R (f;Q') = Var R (f;Q ) + Var R (f;Qg) < sup Var D (f;Q.) + sup Var D (f;Q.) Q 1 R l 1 Q 2 R 2 2 = Var (f) + Var D (f). R 1 R 2 Taking supremum on the left, we get
PAGE 36
33 Var (f) < Var (f) + Var (f). n n R For the other inequality, if Q ,Q ? are any grids on R R respectively, then Q U Q is a grid on R, and we have Var (f;CL) + Var (f;Q.) = Var(f;Q) n 1 n_ d n < sup Var D (f;Q) = Var_(f). _ n n Q=OXT Since this inequality holds for any grid Q on R , we have sup Var (f;Q,) + Var D (f;Q ) < Var(f), Q =oxt 1 2 i.e., Var (f) + Var D (f;Q.) < Var_(f). Similarly, Q being arbitrary, we have on taking supremum for Q . Var D (f) + Var D (f) < Var_(f). R 1 R 2 R Putting the two together, we have the equality: Var R (f) + Var R (f) = Var R (f). iv) Assume R. = [ (s ,t) , (s' ,t ') ] is contained in RÂ„ = [ (q ,r) , (q' ,r ') ]. Then we have q Â£ s < s' i q', and r Â£ t < t' Â£ r' (see Figure 23). By the additivity property, we have Var D (f) = Var D (f) + Var D (f) + Var D (f) + Var D (f) + Var D (f). R 2 R 1 R 3 R U 5 R 6 Since each term on the right is nonnegative, we have Var (f) > Var (f). R 2 R 1
PAGE 37
3U t t R 5
PAGE 38
35 rectangle Rj = [(s't' ) , (s' ,t ' ) ] e Q', and we have I IK f 1 " IKf 11 IK f AÂ„, f 1 R k,l ' \,1 " " \,1 \,1 " = If cÂ» k+1 .t 1+1 )f o k+1 .V^vVi^W^ 3 ^ '^1 e e , e e _ s 4mn 4mn 4mn 4m n mn" Summing up, then, we obtain I I K f E A f  I (A f A f) R eQ i R.eQ' J 00, there is a grid Q with Var r ,,(f;QÂ„) > N + 1. By the above, there exists a grid Q' with Lz ,z J N N rational coordinates such that Var r ,,(f;Q') > N + Â— > N,* hence Lz ,z J N 2 sup Var r ,.(f;Q) =Â• + Â« Var r , n (f). Similarly, if Q=axx [Z ' Z ] [Z ' Z ] Q rational Var r ,,(f) = a < Â°Â°, for every e > there exists Q = oxt such that L z , z J e Var r , n (f;Q ) > a =Â•, and there exists Q' = o'xt' with rational Lz ,z J e 2 E * Again, Q' is not a partition of R, so this must be interpreted directly as the sum given in Definition 2.2.4(a).
PAGE 39
coordinates such that Var r ,,(f;Q ) Var r ,,(f;Q')  < ~, Lz,z J e Lz,z J e ' 2 hence Varv , n (f;Q') > Var r Â„ , n (f;Q ) Â§ > (a Â§) Â§ Lz,z J e lz.zJ e 2 2 2 (f). e arbitrary => sup Q=oxx Qrational Var r ,,(f,Q) = Var r ,[z ,z J [z ,z J An important property of the variation in one dimension is that a function of finite variation f is right continuous if and only if 11m Var r ,,(f) = for all s in the domain of f. Unfortunately, [s,s J s + + s we do not have this equivalence in two dimensions without additional assumptions about f. We do have one implication, however. Theorem 2.2.6 . Let f: R * E be a function with Var_(f) < Â» for every bounded rectangle R. If f is right continuous, then for every 2 z, z , u in R with z < z < u, we have Var r ,,(f) = lira Var r ,(f). [z.z ] u++z , [z,u] (Note: The notation u+ + z' means u Â•* z', u > z'.) Proof . We divide the region outside [z,z'] and inside [z,u] into three parts, labeled R , R , R (see Figure 24). The proof consists R 3
PAGE 40
37 of showing that, as u decreases to z', the variation on each of the three rectangles R , R , and R vanishes. We shall give separate proofs for R and R , and the proof for R is identical to that for R p . Note first of all, that for u' such that z' < u' < u, the corresponding rectangles R", R', R' satisfy R ' CT R , R'CH,, R'CR_, so by Proposition 2.2.5(iv), Var D ,(f) < Var D (f), etc., and 3 3 R 1 R 1 so each of the limits lim Var (f), lim Var D (f), lim Var D (f) *R. Â• K~ , n _ ui4z 1 u + + z 2 u + + z 3 exists and is nonnegative. a) Denote z' = (s',t') u = (p,r). We show that lim Var r . , . ,. . . ,(f) 0. p^s' [(s Â» t Mp.D] r + + t' Assume not: then there exists a>0 such that, for all u>z', we have Var r , ,(f) > a. Let uÂ„ > z', g > 0. Denote u. = (p.,r.). [z ,u] Since Var r , i > a, there exist partitions o Â•. s' = sÂ„ _ < s_ . < [z ,uJ 0,0 0,1 S 0,2 < Â•Â•Â• < S 0,m = P 0' V fc ' = fc 0,0 < fc 0,1 < 'Â•Â• < fc 0,n = r SUCh that i} l A [(s t ) (s t )] fS > a 0* Km US 0,i 0,j '^ S 0,i + 1 ,l 0,j + r J and 0Â£j a Â§. VÂ°0 XT 6 VÂ°0 XT 6 VÂ°0 XT 8 R C I R CII R C III B 6 8 (See Figure 25.) Since each of the three sums is less than the
PAGE 41
33 ''Â•'O,!* U l,l (s',t') (s o,i >r o ) ( PoÂ» r o> ^0^0, 1 } I
PAGE 42
39 Denote u = (p ,r ) (s ,t .). By assumption, Var r , ,(f) > a, etc. Continuing In this manner, we construct a lZ ,u_J sequence u , u , u, with u, > u, , > z' for all i such that, for i 1+1 "0' "1 ' 2 all 1, the total variation of f on the rectangles making up [(s',t'),(p,r)]\ [(s',t'),(p.,r)] is greater than (a ) + (a ) ... + (a *y) ia ,1, *>*' hence we have Var r , .(f) > ia Â£ for all i, i.e., [z ,u Q ] Var r ,(f) Â» + Â», a contradiction of the hypotheses that [z,u ] Var (f) < Â« on every bounded rectangle. Hence, for any a<0, we have R < lim Var r . , ,, , ,,(f) < a => lira Var D (f) = 0. p ++ s' [(s ' l ) ' ( P' r)] u*+z' R 1 This takes care of R . b) We show now that lim Va~ R (f) = 0, or, more precisely, that u++z' 2 lim Var r , , . . , . ,. ,(f ) p^s' C(s M*^'* )] 0. (See Figure 26) . We proceed as before, by contradiction. Assume there exists a>0 such that (s,t') z(s,t) (P,t0 R 2
PAGE 43
40 Var r , , . . , . ,.,(f) > a for all p > 3'. Let, then, p^ > s', let Li.3,t;,ip,t;j e>0. There exist partitions n : s' = s < s < ... < s = d , 0,0 0,1 0,m y 0' T : l = Vo < '0,1 < '" < \n ' t# 5UCh that E ' A R f 5 > Â°" VV T 6 Now, consider the rectangle [(s',t),(s. .,t')]. For each i, u, 1 i = 1,2,..., n, there is r. , s' < r < s n , r < r such that for each i [(3 ''Vii ) '^r t o,i )] f l <Â£ by right continuity at each (s',t ) (r n ,t') (s 01 ,f) (p ,t') t, \ 0,2 '0,nl (s',t) r n r 2 v 1 (s Q1 ,t) (p ,t) Figure 27 Breakdown of o n xt Subdividing each of the leftmost rectangles of xt in this manner creates a refinement F_ of o n XT n , hence I \h f  > o. Now, we also have that R y eP Y n E A i1 C(3 '' t 0,i1 ) '^i' t 0,i )]f ' i = 1 2 i+1 "~ 2 '
PAGE 44
41 n 1 K f l " E S A [(s' t ) (r t )l f > a ' \ * The rectangles in this sum form a partition of the n rectangles [( V t),(p,t 0J )], [(r 2 ,t 0J ),(p,t 0>2 )], ... [(Vt^Mp.t')] (shaded rectangles in Figure 27). Thus n 11 [(r i' t 0,i1 ) Â» ( P' t 0,i )] > E A f E & f > a Â§ R y eP Y 11 L ^ S '^0,11 j,lr i ,t 0,i ; d In particular, then, Var r , . . , . ,., > a % . Let p = r . By L [r ,t ) , (p ,t ) J 2 H n ssumption, Varr, , > ,.,(f) > a, so we can repeat this as; ..US . t i . U n procedure and get another point s' < r' < r such that n n Var r , , , Â» , . M1 (f) > a T7 . Continuing in this manner, we can L(r ,t),(p ,t ) J n construct a sequence p , p , p , such that i) Pj > P l+1 > s, p ++s ii) Var r , .. , k ^i(f) > a frr for alL i > L(p i+1 ,t),(p i ,t )] 2 i+1 We have, then, by additivity, 11 [(p. ,t),(p ,t )] L(p J+1 ,t),(p ,t )] 11 i1 > I (a . . ) = ia E > ia e j0 2 J j0 2 J i1 Then Var (f) > E Var r , , , N ,(f) > iae. R 2 j=Q [(p. +1 ,t),(p.,t )]
PAGE 45
42 e arbitrary => Var (f) > ia for all 1, hence Var (f) = + Â», again a R 2 R 2 contradiction. Hence lira Var r . , , , ,,.(f) Â£ a for any a>0, so P++S ' [(s t).(P.t )] lira Var r . , , . ,.,(f) 0. p++s , [(s ,t),(p,t )] By the same argument, lim Var r , ,_,. , , N ,(f) = (the R_ Â„,,,.C(s,t ),(s ,r)J 3 r + + t case). Putting everything together, we have lim Var r .(f) = lim [Var r , n (f) + Var,, (f) + Var (f) + Var (f)] , , , Lz.u] , , , [z,z ] R R. R_ u++z u++z 1 2 3 = lim Var r ,,(f) + lim Var (f) + lim Var (f) + lim Var n (f) [z,z J RÂ„ RÂ„ R_ u++z u++z 1 u++z 2 u++z+ 3 = Var r .At) + + + 0) = Var r ,,(f), [z,z J [z,z J which is what was to be proved. Remarks . 1) As we stated above, the converse is not true. However, if lim Var r ,(f) = Var, ,,(f), then lim Var r , .(f) = 0, and from u**z' Cz ' u] Cz ' Z ] u**z' [Z Â« U] ' A [z',u] f l " Var [z',u] (f) WS get that lirn ,' A [z',u] f ' = Â°Â» i ' e ** f iS Incrementally right continuous. 2) Returning to Example 2.1.2 , for f as defined there, we have Var f ,,(f) = on any rectangle [z,z'], since I Ja r (f)f = Lz,z J R eP a a Â— I = for any partition P of [z,z'], so lim Var r .(f) = R cP " u**z' [Z ' U] a Â— Var r ,,(f). However, if f is constructed from a function g that is Lz,z J not right continuous, then neither is f. In fact, for all e>0,
PAGE 46
H3 f(s+e,t+e) = g(t+e), so lim f(s+e,t+e) = lira g(t+e) * g(t) = f(s,t), so f is not right continuous. In the next section, we shall give sufficient additional conditions on f for the converse to hold. 2. 3 Functions of Two Variables with Finite Variation As the example in the previous section (at the end) shows, the requirement that Var (f) < Â» on bounded rectangles R is by itself n insufficient to give all the properties necessary to associate a Stieltjes measure to it. In order to deduce properties of f from its variation, we need some extra conditions. It seems natural to require that each of the onedimensional paths also have finite variation, but we do not need quite that much. In fact, if Var (f) < Â» on all bounded R rectangles R C R , and if the onedimensional path f(Â«,t n ): R E has finite variation for some t , then the paths f(Â«,t) have finite variation for all t. More precisely, for any s>0, we have Var [0>s] f(.,t) s] f(.,t ) Var [( Â§ (g ^ ] (f (Note: We replace the second term by Var r .Â„ , . ..(f) if [(0,t) ,(s,t Q ) ] t < t.). To see this, let o: = s^ < s, < ... < s = s be a 1 n partition of [0,s] (Figure 28): We have, for each i, < i < n1, f(s. +1 ,t)f( Si ,t) = f(s i+l ,t)f(s i ,t)f(s. +1 ,t ) + f(s i ,t ) f(B 1+1 ,t )f(8 lf t ) ^ f( Si+1 ,t)f(3 l ,t)f(S l+1 ,t ) + f(8 l ,t ) + l f(s 1+ rVf(s i'Vl'
PAGE 47
H'4 Â—\ 1 Â— S l S 2 'n1 Figure 28 Partition of bounding Var ro ,f(',t) L0,s J Summing over the i's, and denoting R = [ (s ,t ) , (s ,t ) ], we have n1 n1 Z f(S i+1 ,t)f(3.,t)I < E (A R f + f(3 l+1 ,t )f(3 lf t )) n1 n1 Z A f + Z f(s ,t )f(3 ,t ) i=0 i i0 Var C(O f t ),( S ,t)] (f) + Var C0,s] f( 'V ( Â° r Var [(0,t),(3,t )] (f) + Var [0, S ] f( ' t ) lf ts] f(.,t) < Var [(Q Stt)] (f) + Var [0>g] f (. ,t Q ) . By the same proof (using partitions of [0,t]) we see that if the onedimensional path f(s ,Â•): R + > E has finite variation for some
PAGE 48
^5 s , then the paths f(s,Â») have finite variation for all f, and in fact (same proof) Var ro ,f(s,0 S Var r , Â„, , ...(f) + Var ro ,f(s n ,Â«). [0,t] C(s ,0),(s,t)] [0,t] 0' Up to now, we have avoided using the phrase "f has finite variation" because of the weakness of the condition Var (f) < Â°Â° for R K bounded. We shall reserve this term for functions with the additional conditions described above. We will see that this is enough to give the additional properties we need to associate useful measures. 2 Definition 2.3.1 . Let f: R * E be right continuous, with 2 Var_(f ) < Â• for bounded rectangles R C R . We say that f has finite R + variation if the realvalued function f(s,t) = lf(0,0)l + Var [0>s] f(.,0) + Var [0>t] f(0,.) + Var [(0>0)j(Sit)] (f) < 2 for every (s,t) e R . We say f has bounded variation if there exists M>0 such that 2 f  (s,t) < M for all (s,t) g R + . 2 The map f: R * R is called the variation of f . (Note that we use the single bars to distinguish it from the norm in E.) Remarks . 1) Henceforth, the phrase "f has finite variation" will be 2 understood to mean that f(s,t) < Â°Â° for all (s,t) e R . 2) We extend f by outside the first quadrant to get a 2 function defined on all of R .
PAGE 49
3) The "jump at zero," f(0,0), will play a role later, similar to that of the jump at zero in the theory of oneparameter processes. When we associate measures with f, we shall need this term to get some compatibility between these measures and those associated with f. 4) The function f is increasing in both senses of Definition 2.1.3First of all, if z = (3,t), z' = (s',t*), and z < z', then f(s',t') f(s,t) = if (s',t') lies outside the first quadrant; f(s',t') f(s,t) = f(s',t') > if z' > 0, z outside FT. If S z < z', then f(s',t') f(s,t) = f(0,0) Var [Q a ,.f(. f 0) + Var r Â„ , n f(0,) + Var r .Â„ Â„, . , ,,,(f) [0,t'] L(0,0),(s',t')] [f(0,0) Var rn ,f(Â«,0) Var r ,_ .,f(0,Â« 1 ' [0,s] [0,t] +Var [(o,o),(s,t)] (f)] Var r ,,f(,0) + Var r . . o f(0,v [3,3 ] [t,t ] +(Var [(0,0),(s',t')] (f) " Var [(0,0),(s,t)] (f)) > 0. As for the other sense, we have A r ,, f = if z' lies outside the [z,z J ' ' first quadrant. We then deal with the case where S z'. If z is in the third quadrant, i.e., if s<0, t<0, then A r ,,f [z ,z J ' = f(s',t') f(s',t) f(s,t') + f(s,t) = f(s',t') > 0.
PAGE 50
^47 If z is in the fourth quadrant, i.e., if s>0, t<0, then A r z z'] (f) = f (s'.t*) f(s,t') + = If (0.0)  Var [0jS , ] f(,0) Var [ Q ^ #] f (0, Â• ) +Var [(0,0),( S ',t')] (f) " ( l f( Â°' 0) ! + Var ro,s] f( ' 0) Var [0ft#] f(0 f O + Var [(0)0Ms>t , )]( f)) = Var [s,s'] f( ' 0) +Var [(s,0),(s',t')] (f) Â°Â« Similarly, if z is in the second quadrant, i.e., if s<0, t>0, then A r ..(f) = Var r . . , 1 f(0,O + Var r , n . , . fc .x,(f) i0. Lastly, if [z,z ] Lt ,t ] [(0,t) , (s ,t )] Â£ z < z', then we have A (f) = f(s',t') f(s',t) f(s,t') f(s,t) L Z 9 Z J = If (0,0)  Var [0jS . ] f(.,0) Var [0tt<] f(0,.) +Var [(0,0),(s',t')] (f) " ( H f( Â°' 0) l +Var [0,s'] f( ' 0) + Var [0)t] f(0,.) + Var [(0(0)>(s . (t)] (f)) (f(0,0) Var [0i8] f(..0) * Var [0>t<] f(0,.) + Var [(0,0),(s,t')] (f)) + Â« f(0 ' 0) l +Var ro,s] f( '' 0) Var [0ft] f(0,.) + Var [(0>0Ms>t , )]( f) = (Var [(0,0),(s',t')] (f) Var [(0,0),(s',t)] (f)) " (Var [(o,o),(s,t')] (f) " Var C(o,o),(s,t)] (f))
PAGE 51
iJ8 =Var [(0,t),(s'.t')] (n " Var [CO,t),(s,t')] (f) Var C(3,t),(s' f t')] (f) "^ This definition allows us to recover many results analogous to those of functions of one variable, as we show in the next few theorems. Theorem 2.3.2 . Let f: R > E have finite variation f. Then f is right continuous if and only if f is right continuous. Proof . Assume, first, that f is right continuous. We write, for (s.t) > (0,0), f(s,t) f(0,0) Var [0(S] f(.,0) Var [Q>t] f (0, Â• ) Var [((J>0) , (s>t)] (f ) The first term is constant; to show that f(s,t) = lim f(s',t'), it s '+s t'+t suffices to show that i) lim Var f(,0) = Var rn f(.,0) s'++s [0 ' s ] [0 ' 3] ii) lim Var f(0,) = Var f(0,O t'+n L ' J L ' J (s',t')++(s,t) ^Â°'Â°Ms Â»t )] [(0,0), (s.t)] We proved (iii) in Theorem 2.2.6 (taking z = (0,0), z' = (s,t), u = (s*,t')). As for the other two, f right continuous implies f(Â«,0), f(0,Â«) right continuous: in fact, taking t' = t = 0, we have
PAGE 52
149 lim f(s'.t') = lim f(s',0) = f(s,0! (s',t') + (s,t) s' + + s (s',t ')*(s,t) (Recall that the definition of right continuity allows us to take limits along vertical or horizontal paths as well, unlike left limits.) Hence f(Â«,0) is right continuous, so the variation is right continuous, i.e., Var Pn ,f(,0) = lim Var_ ,,f(Â»,0). Similarly, [0,s] s% + 3 [0,s ] taking s' = s = 0, we have f(0,t) = f(s,t) = lim f(s'.t') = lim f(O.t'), (s',t') + (s,t) t' + + t (s',t')*(s,t) so f(0,) is right continuous. The variation is then right continuous, so Var r f(0,O = lim Var r f(0,O. Then each of LU.tj t% + t LU.t J the terms of f is right continuous; hence f is right continuous on R 2 . + Conversely, assume f is right continuous at each point 2 (s,t) e R . Taking t 0, and letting s'++s along the path t = 0, we have Var [0,s'] f( *' 0) " Var [0,s] f( '' 0) = l f l (s 'Â» 0) " f(s,0). In fact,
PAGE 53
50 f(s',0) f(s,0) = f(0,0) + Var [Q g ^f(Â«,0) + Var [Q 0] f(0,) + Var [(0,0),(s',0)] f " ( l f(0 'Â°' +Var [0,3] f( 'Â° : Var [0i0] f(O f O + Var [(o>0)i(g>o)] f) " Var [0,s'] f(> ' 0) " Var [0,s] f( ' 0) Then Var r Â„ _f(Â»,0) = lim Var. ,,f(,0) since Ifl is right CO,s] s , ++s [0,s ] continuous, i.e., the variation of f(Â«,0) is right continuous; hence f(Â»,0) itself is right continuous. A similar computation taking s = shows that f(0,Â«) is right continuous. Then, writing Var [(0,0),(s,t)] (f) = l^ 3 ' ' ' f( Â°' 0) Â« Var [0,s] f( '' 0) Var [0>t] f(0,.), we see that each term of f is right continuous. Now, let (s',t') Â£ (s,t), (s',t') * (s,t). We have f(s',t') f(s,t) = f(s',t') f(s',t) + f(s',t) f(s,t) Â£ f(s',t') f(s",t) + f(s",t) f(s,t). We note now the following inequalities (cf. Figure 29): 1) f(s',t') f(s',t) = f(s',t') f(s',t) f(O.t') + f(0,t) + f(O.t') f(0,t) < f(s',t') f(s',t) f(O.t') + f(0,t)f + f(0,t') f(0,t) 
PAGE 54
51 = ' A [(0,t),(s',t')] f l + I" '*'' f( Â°' t) l Var [(0,t),(s',r')] (f) + I'* '*'' f(0.t). 2) f(s',t) f(s,t) = f(s',t) f(s,t) f(s',0) + f(3,0) + f(s',0) f(s,0)  < f(s',t) f(s,t) f(s',0) + f(a,0) f(s',0) f(s,0) ' ,A [(s,0),(s',t)] (f) l + > f(3 '' 0) f(3 ' 0) B Var [(s,0),(s',t)] (f) + l f(5 '' 0) ?(*>Vl Putting everything together, we have f(s',t') f(s,t) S f(s',t') f(s',t) + f(s',t) f(s,t) * Var [(o,t),(s',t')] (f) + I" '*' 5 " ^'^l + Var ;cs,o),(s',t)] (f) + l f(s '' 0) f(s Â« 0) " = [Var [(0,0) f (s',t')] (f) Var [(0,0),(s,t)] (f)] + f(0,t') f(0,t) + f(s',0) f(a,0) (cf. Figure 29). As we showed above, each of the three terras on the / Figure 29 Rectangles used in (1 ) and (2)
PAGE 55
52 right tends to zero as (s',f) decreases to (s,t) , so we have lim f (s'.t') f(s,t)  = 0, (s ',t ') + (s,t) i.e., f is right continuous. I Remark . We still have the same result if we extend f, f by zero 2 outside R . + The next result concerns the existence of "onesided" limits and "limits at infinity." o Theorem 2.3.3 . a) Let f: R + E be a function with finite variation f. Then each of the following limits exists* at each z = (s,t) e R 2 : 1) f(s +f t + ) = lim f(s',t') s ' + + s t'++t 2) f(s_,t + ) = lim f(s'.t') s' + t s t' + + t 3) f(s_,t_) = lim f(s'.t') s ' + + s t' + tt H) f(3 + ,t_) = lim f(s',t'). s'++s t'+tt Moreover, if f has bounded variation (i.e., if there exists M>0 such that f(s,t) < M for all (s,t) c R^), then each of the following * Of course, on the axes, not all these limits make sense. It will be understood that at each point we take limits from quadrants where f is defined.
PAGE 56
53 "limits of infinity" exist: 1 ') f(s + ,Â«) lira f(s'.t') s '+ + s t 'too 2') f(s_,Â°Â°) = lira f(s'.t') s 't+s t 'too 3') f(*,t ) = lira f(s',t') t'++t s ' + 00 4') f(",t_) = lim f(s',t'), and especially t't+t s 'Â•*Â°Â° 5') f(Â») = lim f(s',t') exists. b) If, moreover, f is right continuous, then the onesided limits along the vertical and horizontal paths f(s,), f(*t) are equal to the following. i) lira f(,t) = lim f(s,) = f(s+,t+) (right limits) s' + + s t'++t ii) lim f(. ,t) = f (s,t + ) , and s '++3 lim f(,t) = f(Â»,t+) if f is bounded, s'+m iii) lim f(s,) = f(s+,t), and t't+t lim f(s,Â«) = f(s+,Â») if f is bounded. t't* p Remarks. 1) Here f is defined on R : if we wish to use an f defined + also on the "boundary at infinity," the above limits will be denoted with the symbol Â»in place of Â°Â°.
PAGE 57
54 2) In general, a function of finite variation can have eight different limits at a point: the four "quadrantal" limits from part (a) of the statement, plus the four onesided limits along the vertical and horizontal paths. Part (b) of the statements says that if F is right continuous, the onesided limits can be incorporated into the quadrantal limits, so there are only four distinct limits at a point (s,t) (at most). The first part of (b) says that both right limits along the vertical and horizontal paths are equal to limit (1), the second part says the left limit along the horizontal path is equal to limit (2), and the third part says the remaining left limit is equal to limit ( H) , giving the division of the plane shown in Figure 210. There are analogous considerations for the "limits at infinity." (s,t) (1) (3) ObO (s.t) (2) (s,t) (4) Figure 210 The four "quadrants"
PAGE 58
55 Proof , a) Assume first that f has finite variation f. The proofs of limits (1 )(*Â•) are similar; we treat (1) first. We shall show that for any sequence (s ,t ) with s ++S , t + + t, the M n' n n n sequence f(s , t )} ,, i s Cauchy in E. We shall do this by denial: n n ncN Let (s ,t ) be a sequence as above and assume that f(s ,t )} Â„ is n n n n neN not Cauchywe shall reach a contradiction. Since f(s ,t ) is not Cauchy, there exists eÂ„>0 and a subsequence n n (n, ), Â„ such that, for all k, we have k keN If (3 ,t ) f(s ,t )  > e n . n, ' n, , n, n. k+1 k+1 k k We will henceforth denote this sequence by f(s ,t )} .,# so we have n n 'neN the inequality for all n. For j given, consider the subdivisions s > s > ... > s. 3 = s of [s.s^ and t 1 > t 2 > ... > t. > t. . = t of [t,t,]. For i = 1,2,...,j1 denote R = [( s ,0) , (s , t ) ] and R' = [(0,t ) ,(s ,t )] (Figure 211). For each i, we have A R f = lÂ«VW ' Â«Â« H .t 1M ) " f ( s i'Â°) + Â«Â» 1+1 .0). so u R n >iÂ«vW ^im'Vi'I ! f(s iÂ« 0) f(s i + r 0) i => f(s.,0) f(s i+1 ,0) A R f > f(s lf t l+1 ) f(8 1+1 ,t 1+1 ). We hav similarly A R ,f j = IfCSj.tj) f( Sl ,t. +1 ) f(0, ti ) f(0,t. +1 ) 1 > Ifts^tj) r(s 1 ,t 1+1 ) Bf(o,t i ) f(o,t i+1 
PAGE 59
56 t . J s s .s , , .1 J 1+1 ) s j+l s i Â•s s 1 I Figure 211 Partition of [ (0, 0) , (s ,t) ] Â•> f(0,t.) f(0.t l + 1 ) A R .f i Ifls^t.) f(s if t i+1 ). Puttlm the two together we have, for each i f(s.,t.) f(s l+l .t l+1 )= If(s.,t.) f( Si ,t. +1 ) f(s.,t i+1 ) f(s l+1 .t l+l ) < !f(s. >ti ) f(S i ,t. +1 )l !f(s.,t. +1 ) f(3 i+1 ,t l+1 ) < A R ffl + A R .f + lf(s.,0) f(s 1 + 1 ,0) f(0,t.) f(0,t. + 1 ) Now, upon summing over the i's, we have
PAGE 60
57 J1 Z f(s.,t ) f(s. ,,t. J " 1 i i + 1 i + 1 J1 l (A R . f ' + 'V fl + l f( V 0) " f(s 1+1 ,o) J i = 1 i + f(o,t i ) f(o,t i+1 )f) j1 j1 E (A f + A_,f) Z f(s.,0) f(s. ,,0) 11 R i R i 11 X 1+1 j1 + Z f(0,t ) f(0,t. ,) 11 i i+1 L (0,0; , (s ,t ) J [s.,s J [t.,t J (since the union of the R , R' is contained in [( 0,0) , (s , t ) ] ) * Var [(o,o), ( s l ,t 1 )] (f) + Var [o.,s/ ( ' 0) + ^[o.t/^ (J1)e , hence If^s^t,) > (Jl)e The left hand side does not depend on j , so letting j Â» Â•, we obtain f(s ,t ) = + Â°Â°, a contradiction on the assumption that f < Â»,
PAGE 61
58 Thus, for any sequence (s ,t ) decreasing to (s,t), the sequence n n f(s ,t )} Â„ is Cauchy in E complete; hence lim f(s ,t ) exists. We n n neN n n show now that we get the same limit for any sequence decreasing to (s,t). Consider two sequences (s ,t ) and (s'.t"), both decreasing to n n n n (s,t). We construct a new sequence (p ,r ) as follows. * n n We set (p.tO = (s ,t ), (p 2 ,r ) = the first term of (s^.t/) smaller than (p ,r ), (p ,r ) = the first term of (s ,t ) after (s ,t ) smaller than (p ,r ), etc. The sequence (p ,r ) then decreases and converges to (s,t) since both the evenand oddnumbered terms do. Then L = lim f(p ,r ) exists from above. Looking n n n at the oddnumbered terms, we have L = lim f(p ' r 2k+1 ^ " But the k '" '' oddnumbered terms form a subsequence of (s ,t ), and we know f(s ,t ) converges, so L = lim f(s ,t ) as well. Similarly, since n n n n n the evennumbered terms form a subsequence of (s',t'), we obtain n n L = lim f(s'.t') as well. The limit is then independent of the n n particular sequence, so lim f(s',t') exists, and it is (s',t')*+(a,t) this limit we denote by f(s ,t ). The proofs of C 2 ) Â— ( M ) are similar, and we will omit some computational details where they are identical to the ones for (1). Proof of (2). We consider a sequence (s ,t ) * (s,t) , with 3 tts n n n t + + t, and show that the sequence ff(s ,t )} ,, is Cauchy in E, which n n n 'neN we again do by denial.
PAGE 62
59 As in the proof of (1), we extract an e > 0, and (s ,t ) such that (s ,t ) + (s,t), and for each n we have s , > s , t , < t , n n n+1 n n+1 n and If (s , ,t ,) f(s ,t ) I > e.. As in (1), for given j, conside1 n+1 n+1 n n " the subdivisions s < s < ... < s. < s = s of [s ,s] and t > t > ... > t. > t = t of [t,t ]. For i = 1,2,...,j1, denote R, = [(s.,0),(s. ,,tj], R; = [(0,t. ,),(s. ,,t.)]. (See Figure 212.) l ii+1i i i+1 i+1 l We have, for each i (similar to before): ' a r f ' = ' f(s i + i'V " r^i^S ~ f(s i+ i' 0) + f(s i' 0) l i > f(S i+1 ,t.) " f(3 1 ,t 1 )l f(S 1+1 ,0) f(s.,0), hence t.
PAGE 63
60 f(s. +1 ,0) f(s lf 0) A R f Z !f(s 1+1 ,t.) f(s.,t.) A R jf If(3. +1 ,t.) f(3. +1 ,t i+i ) f(0,t.) f(0,t l+1 )I * f. f(8 .t ) f(s i + 1 ,t i + 1 ). i Here we diverge from the proof of (1), since the rectangles R., R' 11 overlap (see Figure 212). From the first inequality we have, upon summing over i, J1 J1 I f(a ,t ) f(s ,t ) i I (A f f(s ,0) f(s 0)) 1=1 1*1 i j1 J1 = * K f + E f(s .0) f(s 0)J 1=1 111 v aiÂ°r/^ ^ , ,,(f) + Var r ,f(',0) [(0,0) ,(s,t )r [s ,s] as in the proof of ( 1 ) f(s,t ) (as in the proof of (1)). Also, by a similar computation, we have
PAGE 64
61 j1 Z f(s. ,,t.) f(s. .,t. , 1+1 i 1+1 i+1 i1 j1 J1 < z a ,f + z f(o,t ) f(o,t ) 11 i i=1 [(0,0) ,(s,t. ) J Lt.t.J * f(8,t ). Putting the two together, we have (as in (1)): j1 Z 1 = 1 I !f(S i ,t i ) f(3 i+1 ,t i+1 ) j1 z i1 = Z f(8 ,t ) ft^.V '<Â» 1 + 1'V f(3 i + 1 ,t i + 1 )
PAGE 65
62 The proofs of the last two are the same as those of the first two: to prove (3), we use the same method as that of (1) to get J1 I f(s ,t ) f(s i+1 t i+1 )l f(s,t) (instead of f(s ,t )), and to prove (4) we use the same method as that of (2) to get j1 I f(s ,t ) f (a Â»*Â« +
PAGE 66
63 i+1 ^ R.' 1 S S j " S i + 1 s i Figure 213 Partition for the "limit of infinity" f(s+,Â») J1 I IfCSj.tj) f(3 i+1 ,t i ) $ f(s if t.) < M. Similarly, J1 I Â«B .t ) f(Â» 1+r t ) $ f(s t )
PAGE 67
64 and we get a contradiction as before. Then f(s ,t ) is Cauchy; hence n n lim f(s ,t ) exists, and we prove the limit is the same for any n n n sequence the same way as before. This illustrates the difference between the proofs of (1)(1) and those of (1')(5'): We do the same computation for the limits at infinity, but the value of f turns out to be at a point depending on j, so we further majorize it by M. For (3') we have J1 I f(s.,t ) f(s. .,t.) Â£ f(t .s.) < M ' i i i+1 I " 1 J J1 I f(s i+1 ,t i ) f(a l+l .t l+1 ) S f(t 1>S .) < M, so as before j1 (J1)e < I f(a 1 ,t 1 ) f(s l+l .t l+1 ) S 2M, and we conclude as above. For (2'), (4'), and (5'), we follow the same computation as in the proof of (1) and obtain j1 (j1)e < I IfCSj.tj) f(s i+1 ,t l+1 ) < f  (Sj.tj) < M and conclude as in the proof of (1). This completes the proof of (a). Proof of (b) . Assume, now, that f is right continuous (order sense!). We shall deal with (u) and (ui) first).
PAGE 68
65 Ad (ii) . Denote L = f(s,t+), let z > 0. There exists 5 > such that for all (s',t') with s't, ss ' < 5, t 't < 6, we have f(s',t')L < . Let, then, s' < s with ss' < 5: since f is right continuous, there exists a point (s",t') with s' < s" < s, t' > t, t't < 6, such that f(s',t) f(s",t') < . But we also have " 2 jf(s",t')L <  , hence f(s',t)L < f(s',t) f(s",t') f(s",t')Lfl <  +  = e . Thus, lim f(s',t) = L = f(s,t+). s ' 1 1 s The proof of the limit at infinity is much the same, except that instead of ss' < 6, there is N such that for all s' > N, the conditions hold. We then take s' > N, s" > s, and the remainder is the same. Ad (iii) . The proof of (iii) is the same as that of (ii), with the roles of s and t being reversed: for e > there exists 6 > such that for all s' > s, t* < t, etc. The remainder is the same. Ad (i) . This follows immediately from the definition of right continuity: all three limits are equal to f(s,t). We should remark, however, that it is the same to define right continuity using the open quadrant: the limits along the horizontal and vertical paths are then the same as the "quadrantal" limit. In fact, for e > 0, choose 6 > so that for s' > s, t' > t, s's < <5 , t't < 6, If (s, t)f (s ',t ') J < ^ . 2 Then for any s' > s with s's > 6, there is a similar " Â— neighborhood" for the point (s',t). Pick any point in the intersection of these " neighborhoods," and apply the triangle inequality as before. Our next result concerns the existence of a "Jordan decomposition" for functions of two variables with finite variation:
PAGE 69
66 in two variables, we have two distinct definitions of "increasing," but our decomposition satisfies both. Proposition 2.3.^ . Let f: R * R have finite variation (again, in the sense of Definition 2.3.1). Then we can write f = f f , where f and f are increasing in both senses of Definition 2.1 .3, namely a) for (s.t) Â£ (s',t') we have f (s,t) < f (s'.f) and f (s,t) S f (s'.t') and b) for (s.t) < (s'.f), A r , .. , , Â„, N ,(f,) ^ and L(s,t) , (s ,t ) J 1 [(s,t) , (s ,t )] 2 Proof . For (s,t) e R^, set f (s,t) = f(s,t), fgCs.t) = f (s,t) f(s,t) = f(s,t) f(s,t). In remark (4) following the definition of f (Definition 2.1.3), we showed that f is increasing in both senses. We then have only to deal with f . a) Let (s,t) Â£ (s'.t'). We have f (s'.f) f (s,t) = f(s',t') f(s'.t') (f(s,t) f(s,t)) = (  f  (s'.t ') f(s,t)) (f(s'.t') f(s,t)) We shall show f(s'.t') f(s,t) ^ f(a',t') f(s,t). Denote by R the rectangle [(0,t) , (s'.t ')], by R the rectangle [ ( s,0) , (s ', t) ] (Figure 21*0 . We have f(s'.t') f(s,t) < lf(s',t') f(s,t)
PAGE 70
67 If(s'.t') f(s',t) + f(s',t) f(s,t) Jf(s'.t') f(s',t) f(s',t) f(s,t) = A f + f(O.t') f(0,t) + A . f f(s'.O) f(s,0) 1 2 < A f f(0,t') f(0,t)  A_ f + f(s',0) f(s,0) < Var_ (f) + Var r . . ,,f(0,O Var D (f) + Var r , n f(Â«,0) n. Lwt J Kp LS,SJ = (Var r ,Â„ Â„. . , ...(f) Var r ,Â„ . . n (f)) [(0,0) ,(s ,t ')] . [(0,0) ,(s,t) ] + Var [t.t'] f<0 ' + Var [s,s'] f( '' 0) = f (s'.f) f (s,t) (cf. Remark H following Definition 2.3.1). t"
PAGE 71
66 f (s'.f) f (s,t) = (f(s',f) f(s,t)) (f(s'.t') f(s,t)) > 0, i.e., f (s,t) Â£ f (s',t'), b) Let, (s,t) < (s',t'): we have A [(a i t>,(s\t')]
PAGE 72
CHAPTER III STIELTJES MEASURES ON THE PLANE In this chapter we extend the classical correspondence between functions of finite variation on the real line and Stieltjes measures on the real line to the case of functions and Stieltjes measures on R 2 . 3.1 Measures Associated With Functions 2 Given a function f: R + E with finite variation on bounded 2 rectangles, right continuous (in the order sense!) on R , we can associate a unique measure with finite variation. The statement and proof we give are due essentially to Radu [16]. The statement is a little more general than we really need, but no further difficulties are encountered by this; we also use rightlimits instead of Radus's leftlimits, but this is just a matter of choice. The term "bounded variation" in the statement refers to the variation of f on rectangles as in Definition 2. 2. H; as we have seen, this is weaker than the requirement that f be bounded. 2 Theorem 3.1.1 (Radu). If the function f: R + E is of bounded variation on R 2 and if the right limit f(s+,t+) (of. Theorem V.3.3 for 2 definition) exists at each point (s,t) of R , then there exists a 2 Stieltjes measure m on R with values in E, uniquely determined, with finite variation, and such that for all rectangles 69
PAGE 73
70 R = ((s,t) , (s ',t ')] we have m(R) = f(s'+,t'+) f(s'+,t+) f(a+,t'+) + f(s+,t+) V f .>Proof . We give the proof in several steps. 2 1) Let 6 be the family of rectangles of R" of the form (z,z'] t z
PAGE 74
71 Denote R = (s,s'] x (t,t'] R 2 = (s,s'] x (t',t"l (Figure 3 _ 1). (The proof is the same if R is of the forr (s',s"] x (t,t'].) We have m(R 1 UR 2 ) = f(s"+,t"+) f(s+,t"+) f(s'+,t+) + f(s+,t + ) = [f(s'+,t"+) f(s+,t M +) f(s'+,t"+) + f(s+,t'+)] + [f(s'+,t'+) f(s+,t'+) f(s'+,t+) + f(s+,t+)] (we added and subtracted f(s'+,t'+) and f(s+,t'+)) = m(R ) + m(R ). 3) o has finite variation on 6. We prove this by contradiction; suppose there exists a rectangle J e 6 such that o(J) = (a denotes the variation of o). Then, for each a>0, there exists a finite family (J ), h = 1,2, ...n of disjoint rectangles from 6, J C j for all h, such that h n n I o(J )  > a, i.e., I A (f+)  > a. h1 n h1 n Denote J = ( (s. ,t. ) ,(s ',t ')] for all h. We may, of course, assume h h h h h n that J = \^J J (so that [^J j c 5). Let e > 0. Since f has n=1 h
PAGE 75
72 rightlimits everywhere, there exists a number p > common to all the vertices of all the J, such that h f f(s h+P , V p)i <. h h= 1 closed, bounded rectangle in R , and the family P_ = jl : h = 1...n} forms a partition of I according to Definition 2.2.1. Also for each h, we have A (f+) A (f) h h f(s'+,t'+) f(s'+,t +) f(s +,t'+) + f(s +,t +) 1 v h ' h h ' h h ' h h * h (fCSjJ+p.tjJ+p) f(s^+p,t h +p) f(s h +p,t^+p) + f(3 h +p,t h +p)) (f(s^,t h +) f(s^+p,t^+p)) (f(s h '+,t h +) f(s^+p,t h +p)) (f(S h +,t^) f(S h + p,t^p)) + (f(S h +,t h +) f(3 h + p,t h +p)) ,f(s h +,t h +) " f(s h +p>t h +p) l * l f(s h +>t h +) " f(s h ' +p 'V p) l + ,f(s h +,t h +) ' f ^v p,t h +p) l + l f(s h +,t h +) " f( V p *V p)1 e e e e + + r + ^=r
PAGE 76
73 In particular, we have A. (f) > A (f+)[ E. h h Upon summing over h, we obtain Var (f;P) = I A (f) > I lA (f + ) nc > a ne. h h h Now, denote J = ( (p,r) , (p ' ,r ') ] : we can take p<1 , and decreasing with E, so for any e, we have {J I = IC:[(p,r),(p' + 1,r' + 1)]. Denoting h this latter rectangle by K, we have K Dl for all I (in general, I depends on e) , so Var (f) > Var (f) > a nc . c arbitrary => K 1 Var (f) > a. Now, the collection J depends on a, but they all have union equal to J, so we can repeat the above procedure for any a and keep I C K. Thus, Var (f) > a for any a => Var ..(f) = + Â•, a K K contradiction on our assumption of finite variation of f. Then o has finite variation on 6. 4) o is inner regular on 6. We observe first that, f^om the 2 fact that the rightlimit of f exists at each point of R , it follows 2 2 that for each z e R , e > 0, there exists z e R with z0 such that for any points u,v>z, with  uz I f(z+) f(z') <  . Then f( u +) f(z+) S f(u+) f(z') + f(z') f(z+) $ Z + I = e.
PAGE 77
74 Now, let J e 6, J = ( (s, t ) , (s ', t ') ] with s0. There exists a point (p,r) e J such that for any point (h,k) with (s,t) < (h,k) < (p,r) we have f(h+,k+) f(s+,t+) <  , f(s+,t'+) f(h+,t'+) <  , f(s'+,t + ) f(a'+,k+) <  , as in Figure 32. (We can do this for each by the above, and we use a common n in choosing our (p,r).)
PAGE 78
75 (f(s'+,t'+) f(h+,t'+) f(s'+,k+) + f(h+,k+) jf(s'+,t'+) f(s'+,t'+) + f(h+,t'+) f(s+,t'+) Jf(s'+,k+) f(s'+,t+) + f(h+,k+) f(s+,t+)
PAGE 79
75 6) Since o is inner regular on 6, o is inner regular on C = R(<5) [6, Corollary to Prop. 7, p. 308]. We now show o is regular on C. This follows immediately from the following proposition [6, p. 306]: Suppose that the ring C satisfies the following condition: for every set A e C there exists a set A' e C such that ACInt(A'). Then a measure m is regular on C if and only if m is inner regular on C. We need to show that C satisfies the condition. Let A e C, n then A = \^_J A , A e 6 disjoint. For each i, denote i = 1 1 l A. = C(s 1 ,t 1 ),(s i ',tp]. Then B { = ( (s^l ,^1 ) , (s >1 ,t >1 ) ] belongs to 5: clearly A C Int B for each i; hence n n n A = \^J A C \^J Int B G Int ([^J B.) = Int B, denoting i=1 i=1 i=1 x B = [J B. t C. Take A' = B. We have now an extension o of o to C satisfying: 1 ) o is additive on C 2) o is regular on C 3) o has finite variation on C. Then, by a standard theorem of measure theory, o can be extended uniquely to a Borel measure m of finite variation. This measure clearly coincides with o on 6 , so the theorem is proved. I Remarks . 1) The theorem proved by Radu is for R : we have restricted ourselves to R to enhance the clarity of the proof, but R n presents no additional difficulties (except with notation!!).
PAGE 80
77 2) The theorem holds in particular for the situation we use: that where f is defined on R , with f bounded, and f extended by zero outside the first quadrant. 2 Suppose, now, that f is defined on R , right continuous, with f finite, and extend f by zero outside the first quadrant. As an 2 exercise, we shall compute explicitly the measure of some sets in R using the limits developed in Theorem 2.3.3. More precisely, we will compute the measure of points, intervals, and some rectangles in terms of the "quandrantal" limits of f. 2 i) Let (s,t) e R . Denoting by m the measure associated with f, we compute m ({(s,t)}). We can write (s,t)} = I ) A , n=1 where A denotes the rectangle ((p ,q ),(p',q')l with n Â° F n ,M n F n ,M n (p ,q ) < (s,t) < (p',q'), p + + s, q + + t, p'+s, q'+t (see Figure n n *n n n n n n 3~3). If we decompose A into four parts, labeled IIV in Figure 33, we see that as (p',q') Â•* (s,t) , parts I, II, and IV vanish. We may, n n
PAGE 81
78 therefore, consider the upper corner of A to be (s,t) for each n, so n that we can take A = ((p ,q ),(s,t)] without loss of generality. We n n n have, then, by oadditivity of m , m.({(s,t)}) = lim m (A ) = lim A, (f) i f n A n n n = lim (f(s,t) f(p ,t) f(s,q ) + f(p ,q )) n n n n n = lim f(s,t) lim f(p ,t) lim f(s,q ) + lim f(p ,q ) n n n n n n n n since the individual limits exist by Theorem 2.3.3 = f(s,t) f(s_,t + ) f(s + ,t_) + f(s_,t_). If we note that by right continuity we have f(s,t) = f(s+,t+), we see that the measure of a point is analogous to the measure of a "halfopen" rectangle, except we use the four limits to compute the measure of a point. ii) We next compute the measure m of intervals of the forms: Is} x (t,t'], {s} x [t.f], { s } x [t,t') t {s} x (t,t'), and the analogous "horizontal" intervals. We begin with the closed interval I = {s} x [t,t']. We have I = (1 R , where R are rectangles of the form ((s ,t ),(s',t')l , n n n n n n n=1 with (s ,t ) ++ (s,t), (s',t') + (s,t'). As before, we can take n n n n (s',t') = (s,f) for all n, so that R = ((s ,t ),(s,t')] with n n n n n s < s, t < t, (s ,t ) ++ (s,t) (see Figure 3"^). n n n n
PAGE 82
79 (s n ,to i (sÂ„,t J (s.tO (s,t) r.,t n ) Figure 34 Approximation of an interval by rectangles We have, then, m,(I) * lim m.(R ) I in n lim(f(s,t') f(s ,t') f(s,t ) + f(s ,t )) n n n n f(s.t') lim f(s ,t') lim f(s,t ) + lim f(s ,t ) f(s.t') f(s_,t;) f(s + ,t_) + f(s_,t_) Similarly, we can represent the interval J = [s,s'] x {t} 13 J = () R , where R ((3 ,t ), (s'.t)], a < s, t
PAGE 83
80 (s n' t} (s,t) (s',t) Figure 3 _ 5 Approximation of a horizontal interval Again, we have mÂ„(J) = 11m mÂ„(R ) f f n n lim (f(s',t) f(s',t ) f(s ,t) + f(s ,t )) n n n n n = f(s',t) f(s + ',t_) f(s_,t + ) + f(s_,t_). With these in hand, we can compute the measure of the halfopen and open intervals: We write { s } x (t,t'] I \ {(s,t)}, so m f ({s}x(t,t']) = m f (I) m f ((s,t)}) f(s.t') f(s_,tp f(s + ,t_) + f(s_,t_) (f(s,t)
PAGE 84
81 f(s ,t ) f(s ,t ) + f(s ,t )) f(s.t') f(s_,t^) f(s,t) + f(s_,t ). Similarly, s} x [t,t') = I \ {(s,t)}, so that m f ({s} x [t,t')) = m (I) m f ({(s,t')}) = f(s.t') f(s_,t + ') f(s + ,t_) + f(s_,t_) (f(s.t') f(s + ,0 f(s_,t^) + f(s_,t;>) = f(s +t O f(s_,t;) f(s + ,t_) + f(s_,t_). As for the open interval, we have {s} x (t,t') = s} x [t,t') \ l(s,t)}, so m f ({s}x(t,t')) = m f ({s}x[t,t')) m f ({(s,t)}) = f(s + ,t;) f(s_,0 f(s + ,t_) + f(s_,t_) (f(s,t) f(s + ,t_) f(s_,t + ) + f(s_,t_)) f(s + ,0 f(s_,t_') f(s,t) + f(s_,t + ). We use the same method for the intervals with t fixed. We give the results: m ([s,s')x{tj) = f(s^,t+) f(s^,t_) f(s_,t + ) + f(s_,t_)
PAGE 85
82 m f ((s,s']x{t}) = f(s',t) f(s + ',t_) f(s,t) + f(s + ,t_) m f ((s,s')xft}) Â» T(s'_,t + ) f(s_',t_) f(s,t) + f(s ,t_). iii) We now compute the measure of certain rectangles in R If we allow the possibility of each side being open or closed, this gives 16 different rectangles, and we do not give explicit computations for them all. We shall go into detail for only a few, and indicate the procedure for the remainder. First, we shall give the measure of an open rectangle R = (s,s')x(t,t'). We write R ( )r , where R = ((s ,t ),(s',t')] V n n n n n n n with (s .t ) + (s,t), (s ,t ) +* (s'.f) (see Figure 36). n n n n r l_ _ (s,t) (s ,t ) "1 (s',tO (s'.t 7 ) Figure 3~6 Approximation of open rectangle We have, then, in (R) = lim m fR ) f f n n lim (f(s'.t') f(s',t ) f(s ,t') + f(s ,t )) n n n n n n n n
PAGE 86
83 lim f(s'.t') lim f(s',t ) lim f(s ,t') + lim f(s ,t ) nn nn nn n n n n n n lim f(s'.t') lim f(s',t ) lim f(s ,t') + lim f(s ,t ) '.. Â» n n , , n n n n n n s t+3 3 t+3 s +s s +3 n n n n t'ttt' n t n n t't + 1 n t +t n f(s'.t') f(3',t ) f(s ,t') + f(s,t) We next compute the measure ra f of a closed rectangle R = [(s,t),(s',t')] = [s,s'3 x [t,t']. We can write R =C > \R , where n n R n = ((s ,t ),(s',t')] with (s ,t ) + (s,t), (s',t') + (s'.t') (see Figure 3"7). As before, when n * Â», the rectangles labeled IIII vanish, and so by oadditivity of m_ we can take (s'.t') = (s'.t') f n n without loss of generality. (s',0 n' rr Figure 3~7 Approximation of closed rectangle
PAGE 87
8^ We have m_(R) = lim m_(R ) i in n = lim(f(s',t') f(s ,t') f(s',t ) + f(s ,t ) _ n n n n = f(s',t') lim f(s ,t') lim f(s',t ) + lim f(s ,t ) n n n t t+t n f(s'.t') f(s ,t') f(s',t ) + f(s ,t ). With these in hand, to obtain the measure of other rectangles it is simply a matter of adding or subtracting the appropriate intervals that comprise the sides of the rectangle. We illustrate this procedure (as well as check our work!) by using R and some intervals to compute m ( ( (s,t) , (s ',t ')]). (Note that the rectangle ((s,t),(s',t ')] = (s,s'] x (t,t'].) We have, denoting this rectangle by A (Figure 38), (s,t') i (s,t) (s',f) 1s',t) Figure 38 The rectangle ( s,s ']x( t , t ']
PAGE 88
A = f([s,s']x[t,t']) \ C{s}x[t,t'])} \ {(s,s']x{t}}. (Note that the points (s,t ') , (s ', t) do not belong to A!) Then m f (A) m f ([s,s']x[t,t']) m ( {s}x[t,t ']) m ( (s,s ']x t) ) = f(s'.t') f(s_,t + ') f(s + ',t_) + f(s_,t_) (f(s.t') f(3_,t + ') f(s f ,t_) + f(s_,t_)) (f(s',t) f(s + ',t_) Â•(s,t) + f(s + ,t_)) = f(s'.t') f(s.t') f(s',t) + f(s,t), which is how m (A) was originally defined. The measure of other rectangles can be computed similarly using the parts already explicitly given. 3.2 Functions Associated With Measures In this section we consider the converse problem, namely, given o an Evalued measure m on R with finite variation, is it possible to associate a function with finite variation such that m = m in the f sense of Theorem 3.1.1? The following theorem provides a partial answer to this question. 2 Theorem 3.2.1 . Let m: B(R ) * E be a measure with finite variation i i ? m. There exists a function f: R > E with Var D (f) < on bounded R
PAGE 89
86 rectangles R such that m is the measure associated with f by Theorem 3.1.1, i.e., such that for all rectangles R = ( (s,t) , (s ' ,t ') ] we hav= m(R) = A (f) = f(s'.t') f(s',t) f(s.t') + f(s,t). Proof . Define f: R 2 E by f(s,t) = m( (Â», (s,t)])Â« for 2 (s,t) e R . We show first of all that A D (f) = m(R) for bounded R rectangles R. We have, denoting R = ( (s, t) , (s ',t ')] : A R (f) = f(s'.t') f(s',t) (f(s.t') f(s,t)) = m(( Â— ,(s',t')]) m(( Â— ,(s',t)]) [m(( Â— f (s,t')]) m(( Â— ,(s,t)])] = b(( Â— ,(s',t')] \ ( Â— ,(s' f t)]) m(( Â— ,(s,t')] \ ( Â— ,(s,t)]) = b({( Â— ,(s',t')] \ ( Â— ,(s',t)]} \ {( Â— ,(s,t')] \ ( Â— ,(s,t)]}) = m(R). We now show that f has finite variation on bounded rectangles, i.e., that Var (f) < Â» for bounded rectangles R = [ (s,t) , (s ',t ') ] . Assume note: suppose there exists R = [ ( s, t) , (s ',t ') ] such that Var (f) = + Â°Â°. Denote by R the halfopen rectangle ( (s,t) , (s'.t ')]. n Let o: s = s < s < ... < s = s ' be a partition of [s,s'], t: t Â» t. < t, < ... < t t ' be a partition of [t.t'L and let 1 n * ' Â£ = oxt be the corresponding partition of R (cf. Prop. 2.2.5). Note : ( Â— ,(s,t)] = {z e R 2 : z < (s,t)
PAGE 90
37 Now, for any a>0, there exists such a partition _P such that Var (f;P) > a , i.e., E [a (f) > a . 0 a . a arbitrary => m(R) = + Â», a contradiction since m has finite variation. Hence Var (f) < Â» for R bounded, and R the theorem is proved. I Remarks . 1) We have said nothing about uniqueness of f. In the case of functions on the line, f is determined within a constant by m (i.e., any other associated function g is determined by adding or subtracting a constant from f ) , but here this is not the case. In fact, as we have seen before (Example 2.1.2) that many completely unrelated functions can have zero as its associated measure. 2) The oadditivity of m implies that f is incrementally right continuous, but as we have seen in Chapter II this is insufficient to imply right continuity in the order sense without imposing finite variation on the onedimensional paths f(s;Â«) and f(,t).
PAGE 91
We return now to the situation with f defined on R . right continuous, with finite variation f, both extended by zero outside the first quadrant. We have an important equality we shall make use of in the next chapter, which is given in the following. Theorem 3.2.2 . Let m be the Evalued measure associated with f, and let m  be the realvalued measure associated with f. Then mÂ„ has finite variation m  and we have the equality m f = ivProof . We showed in Theorem 3.1.1 that m has finite variation. The real thing to be proved here is the equality. Let S be the semiring of rectangles of the form R = ( (s,t) , (s ',t ')]. We shall show first that m, = m  on S. We must consider various cases. First of all, if (s',t') < (0,0), then m, ,(R) = m  (R) = 0. We will assume, then, in what follows, that (s'.t') lies in the first quadrant. There are four cases, according to what quadrant (s,t) lies in. 1) Assume (s,t) lies in the first quadrant. Let o: s = s,_ < s, < ... < s = s ' be a partition of [s.s'l, t: t = t < t 1 m ' ' 1 < ... < t = t ' be a partition of [t,t']. Denote P = oxt the corresponding partition of R: P = R. .R. . = C(s.,t,),(s. ,,t. J], < i < m1, < j < n1 Denote by R. . the corresponding halfopen rectangles
PAGE 92
89 ( (s. ,t ) ,(s. + 1 ,t )]. (We shall use this notation in the other cases as well. ) We have E m (R ) = E A (f) < Var (f) = A_(f) i,j ,J l.j H i,j R R (cf. Remark 4 following Defn. 2.3.1), and the right hand side is just mi f i(R). The family (R .) forms a disjoint cover of R with {J R = R, so taking supremum we obtain m (R) < m, ,(R). For the 1,J 1,J J LU other inequality, let e>0. There exists a grill oxt such that E A (f) > Var (f) e = m H (R) e. I.J i,J ! ' But the left hand side is equal to Z m (R ). and the R i,j f iJ iJ forms a decomposition of R, so we have m f (R) > Z m f (R 1 ) = Z A R (f) > m f (R) e , l.j ' ? 1,J i.j i.e., m (R) > m. ,(R) e. Letting e Â» (neither side now depends on the corresponding grill), we obtain m (R)  > m, ,(R); hence m f (R)  = m, f ,(R). 2) Suppose now (s,t) lies in the second quadrant, i.e., s<0, tSO. For any grill oxt, we can refine o by including zero if it is not already there, so that we may take oxt with zero included in o, and compute variations with these grills (Figure 39). Denote by k the index where s = 0: For i < k 2, m,(R, .) = 0; w e also have * 1 f i , j '
PAGE 93
QO (s',f) Figure 3 _ 9 The grid oxt K'Vij'I = rCB kt j*i^ " f( VV " ''VrVi 5 + f( Vi'Vl = f(0,t j+1 ) f(0,t j ). Putting everything together, we get n1 I m (R ) = I f(0,t. +1 ) f(0,t,) + Z A B (f)I l.j f 1,J j = 1 J 1 J i>k R i,j SVar^^ftO.O + VaP [(0>t)((s . tt , )](n = m, f , (R), since m. f i(R) = f(s',t') f(s',t) f(s,t') + f(s,t) = f(s',t') f(s',t) = [f(0,0)I Var rQ s . ] f(',0) Var [Q t . ] f(0,O + Var [(0,0),(s',t')] (f)] " Uf(0.0) Var^^fC.O) Var [0>t] f ( 0, . ) + Var C(0,0),(s',t)] (f)] " [Var [(0,0),(s' ( t')] (f) " Var [(0,0),(s',t)) (f)]
PAGE 94
91 + [Var [0,t'3 f(0 'Â° ' Var CO,t] f(0 '' )] Var, [(o,t),(3',t')3 (f) + Var r t t .'i f (Â°.')Taking supremum over grills oxt on both sides, we get (as before) m (R)  < m, ,(R). On the other hand, for any c>0, there exists a partition t ' of n1 Ct.t'] such that z f(0,t ) f(0,t.) > Var r ,,f(0,O Â§ , and * _g J i J L t , t J 2 a grill o q xt of [0,s ']x[t, t '] such that ia (f) > i , J Var T(0 t) (s' t')1^ " 2 ' We ctloose a common refinement x of t' and i , and extend arbitrarily to get a partition o of [s,s']. Then, for the grill oxt, we have n1 Â£ m (R ) = I f(0,t ) f(0,t ) I a (f) (as before) i.J ' J0 J J o q xt K i,j >Var [t,t'] f( Â°'* ! +Var [(0,t),(s',t')] (f) "I = m, , (R) e. Since the left hand side is bour ded above by m (R) for any grill, we have m f (R) > m . f , (R) e. Letting z * again, we get m (R) 2, m, ,(R), hence equality. The next case proceeds similarly. 3) (s,t) in the fourth quadrant: sÂ£0, t<0. This time, we refine t by including zero if necessary, and compute variations along such grills. Denoting t = a: before, we have (same computation as before) :
PAGE 95
92 m1 Â£ h^CR, .) f (R ifj ) = l f(3 1+1 .0) f(s.,0) I A (f) i=0 i>0 ' i ,j j^k < Var r , n f(0,O + Var r , ., . , ...(f) Ls,s ] [(s,0) , (s ,t )] m, f ,(R) (same as before). Taking supremum we get m (R) < m, ,(R). The proof of the other inequality is the same as that for case (2). 4) Finally, assume (s,t) < (0,0). We proceed similar to the above, but this time we add zero to both o and t, and use these partitions in our figuring of variations (Figure 310). Denote s k = 0, t : = 0. For i < k 2 or j < 1 2, we have m f (R. .) = 0. (s',t*) Figure 310 The grid axx For i = k 1, j = 1 1, we have fm f (R. ) = f(0,0), for
PAGE 96
9? i = k 1 , j > 1, we have m f (R. ) = f(0,t ) f (0,t ) . For j = 1 1, i > k we have m f (R. ) = f(s i+1 ,0) f(s.,0), all as before. Putting everything together, we have m1 I m (R .) = f(0,0) + I f(s. ,,0) f(s.,0) 1,J ' J i = k 1+1 1 n1 I f(0,t ) f(0,t ) I A (f) Jl J 1 J i^k R i,j JSl < f(0,0) Var^^fC.O) + Var^^^fCO,) + Var [(0,0),(s',t')] (f) = f(s',t') = m, f ,(R). Taking supremum again, we obtain m (R) < m, ,(R). The proof the other direction is similar to the ones before: for e>0, we choose a common o,t so that Ef(s 1+1 ,0) f(s 1>0 ) > Var^^jfC.O)  lf(0.t J+1 ) f(0,t.) >Var [0)t , ]f (0,.)  , ^ o lJV^(f)>Var [(0>0)f(s , it , )]( f)Â£. We extend o and t arbitrarily to partitions o'.t' of [s,s']
PAGE 97
94 and [t,t']Â» respectively. The same computation as before gives * IV R i 1>l = l f(0 'Â°H + Sf(3.. 1f 0) f(3,,0) Xf(0,t. + 1 ) iÂ»J a l x T J f(0,t ) I A R (f) axx l,j > lÂ«o.o) + vÂ«. [0i8 , ] f(. i o)f *vÂ«.  + Var [(0,0),(s',t')] (f) "I f(s',t') e = m. f .(R) e. Hence m f (R) > m, (R) c. Letting e 0, we obtain m (R)  > m, .(R); hence equality. This takes care of all the possibilities, so we have m I = m, , 1 f 1 f of S. Moreover, both are oadditive on S; the first since m is by Theorem 3.1.1, the second since f is right continuous by Theorem 2.3.2. Since m , m, , are equal and oadditive on S, they are equal on
PAGE 98
CHAPTER IV VECTORVALUED PROCESSES WITH FINITE VARIATION An important part of the general theory of processes in one parameter is the correspondence between processes of finite variation and measures on R xft (see for example Dellacherie and Meyer [5, VI. 6489] and also Kussmaul [10]). This correspondence finds applications in the notion of dual projections of processes, which are used in the theory of potentials and in decomposition of supermartingales (see for example Dellacherie and Meyer [5, nos. VI. 71113], also Rao [17] and Metivier [11]). In the oneparameter case, the extension of the correspondence to Banachvalued processes is done in Dellacherie and Meyer [5]. In two parameters, the correspondence for realvalued processes is stated (more or less) in Meyer [12]; we shall presently give a more directly applicable (for our purposes) version, along with a proof, as the case of finite variation on R is more delicate (as we have seen). In fact, many times, in the literature results are given for increasing processes, and then extended by defining a process of finite variation as a difference of two increasing processes. The method we use here is a little more constructive. 95
PAGE 99
% 4.1 Definitions and Preliminaries Throughout this chapter, (n,F,P) will denote a complete probability space, (F ) a filtration of subofields of F Z z E R 2 + satisfying the usual conditions. We also assume (F ) satisfies the z axiom (F4) of Cairoli and Walsh [2] (see section 1.2). Throughout this chapter we shall denote by M the product ofield B(R 2 )xF. We now state some definitions we will use in this chapter. (Some are restatements from Chapter I, but we will give them again here for completeness. ) Definition 4.1 .1 . a) A (twoparameter) stochastic function is a function (not p necessarily Mmeasurable) X defined on R xft. Here, X will have values + in a Banach space, usually either in a Bspace E, or in the space L(E,F) of continuous linear maps from E into another Banach space F. We will consider X extended by zero outside the first quadrant, as we p did for functions defined on R . + b) A (twoparameter) stochastic process is a function 2 ? X: R + xtt E, measurable with respect to M = B(R )xF. A process X is called adapted if X : n + E is F measurable for each z e R (see Millet and Sucheston [13] and Chevalier [3] for related notions). We generally use the term raw or brut to refer to a process that is not necessarily adapted, i.e., such that X is Fmeasurable for each z 2 z e R . + p For fixed wen, the map X^(w): R * E is called a path of the process. Each path is a function defined on the first quadrant, so we
PAGE 100
97 shall use the results from the earlier chapters in studying these processes. In particular, the variation of a process is defined in terms of its paths. We have the following definitions. Definition 4.1.2 . a) Let X be a raw process. We call X a process of finite variation if, for each w, the path X # (w): R 2 > R is a function of finite variation in the sense of Definition 2.3.1. We define, for X a process of finite variation, a realvalued process x, called the variation of X by the following: 2 for we n, z = (s,t) e R , X (w) = X > (w)(s,t) x (0t(J) (w) Var [0(S] x > (w)(.,0) + Var [Q>t] X. (w)  (0, Â•) + Var [(o,o M s,t)] ( l x Â»l>. b) If the random variable Ixl = lim Ixl, . < + Â» (which s +. '(s.t) exists since x is increasing in the order sense) is Pintegrable, we say X has lntegrable variation . In this chapter, we shall concern ourselves with processes of integrable variation. We will consider them extended by zero outside 2 R , as we did for functions earlier. Remark . In the book by Dellacherie and Meyer [5], processes of finite variation are defined as differences of Increasing processes. In two parameters, it seems we might have a problem with this, as we have two
PAGE 101
98 distinct definitions of "increasing." However, we have shown (Prop. 2.3. 4) that a process of finite variation, as we have defined it here, can be written as a difference of two processes (apply Prop. 2.3.4 to each path) that are increasing in both senses, thus removing the ambiguity. We give now one more result concerning functions, which will be used extensively in later theorems. Proposition ^.1 .3 . If g: R + L(E,F) is a function with finite variation g (Defn. 2.3.1), then for every x e E and z e F', the 2 2 functions gx: R * F and : R * R have finite variations gx and . (For the realvalued functions we shall use double bars for the absolute value to avoid confusion.) Moreover, if f: R * R + is dgintegrable (i.e., d  g integrable) on a set ICR , then f is d(gx)and d integrable on I, and we have (43.1) x /jfdg = Jjfxdg = JjfdCgx) and also (43.2) = = / I fd. 2 Proof . For the first assertion, let z = (s,t) e R . We have, from Definition 2.3.1 , g(s,t) = g(0,0) + Var [Q a] g(,0) + Var f Q fc] g(0,O + Var (a) < <*> a c(o,o), (s,t)y s)
PAGE 102
99 We have (gx)(0,0) = g(0,0)x < flg(O.O)  x , and from the onedimensional case proved in Dinculeanu [7], we have Var [0,s] (gX)( '' 0) N ' Var [0,3] g(, 'Â° ); Var [0tt] (gx)(0,.) < x Â• Var [0jt] g(0,.). In fact (we prove the first; the proof of the second is identical) for = s < s < ... < s = s a partition of [0,s], we have (gx)(a 1+1 .0) (gx)(3 i ,0) = (g(s. +i ,0) g(s ,0))x * 8(s l+1 ,0) g(5.,0)Ixl for i=0, 1 , 2, . . . ,n1 . Summing over i, we obtain n1 n1 I (gx)(s 0) (gx)(s 0) Â£ I xg(s. .0) g(s.,0) 1=0 1 ] 1 i=0 * 1 l n1 Il x l l 3(s i+1 ,0) g(s 0) 1 = x " i X Var [0,s] s( *' 0) Taking supremum over partitions of [0,s], we get Var ,(gx)(,0) L U f S J S x l' Var [0>s] g(Â»0). The same proof gives Var [Q t] (gx)(0,) Â£ xÂ«Var r t .g(0,Â«). We obtain a similar inequality for the remaining term of gx: For any rectangle R = [(p,q) , (p',q ') ] C R 2 we have A (gx) = (gx)(p',q') (gx)(p',q) (gx)(p,q') + (gx)(p,q)
PAGE 103
100 [g(p',q') " g(p',q) g(p.q') + g(p,q)]x (A p g)x * MBy Then, for any grill axi on [( 0,0) , (s, t) ] , we have 1 K( S t ) (s t )i (gx) ' " l! x B * K (s) lx ' Var [(0,0),(s,t)] (g) ' Taking supremum over grills of [ (0,0) , (s,t) ] , we obtain Var [(0,0),(s,t)] (gx) " X 'Var [(0,0),(s,t)] (g) Putting everything together, we get gx(s,t) = (gx)(0,0)[ + Var [Q a] (gx)(.,0) + Var [Q t] (gx)(0,) + Var [(0,0),(s,t)] (gx) S xg(0,0) x.Var [0jg] g(.,0) + x.Var [0>t] g(0, Â• ) + " X " Var [(0,0),(s,t)] (g) Â« xg(s,t), so gx(s,t) Â£ flxg(s,t) < Â« for all (s,t) e R 2 , i.e., gx has finite variation. The same argument works for : in fact, for (p,q) e R 2 , we have (p,q) = < f (gx) (p,q) J z[ g(pÂ»q) I [x Jz J . The same proof as above then gives
PAGE 104
101 (s,t) Â£ xzg(s,t) < for all (s,t) e R + , i.e., has finite variation for all x e E, z e F'. Now to prove equalities (4.3.1) and (4.3.2): Let I e B(R 2 ), 2 f: R + R be dgintegrable. Assume g(I) < Â°Â°. We shall use the monotone class theorem 1.5.2 to prove the equalities for bounded f, then extended to fintegrable. Let H denote the set of bounded, realvalued, dgintegrable (on I) functions f satisfying (4.3.1). Then: i) H is a vector space (evidently). ii) H contains the constants: let f = a constant. Then x/jfdg = xjjctdg = x/aljdg = x(ag(D) = (ax)g(I) (g(I) is the measure of I for the measure dg), Jjfxdg = (ctx)1 dg = (ax)'g(I) , and /jfd(gx) = JaljdCgx) = a(gx)(I) = a(g(I)x) = (ax)g(I) . Hence xj^fdg = J^fxdg = / fd(gx), which is (4.3.1). iii) H is closed under uniform convergence: suppose f * f n uniformly and (4.3.1) holds for each f . Then n xLf dg * xLfdg, since by Lebesgue dominated convergence we have that f is dgintegrable over I and Lf df Â•Â» f fde. ; I n J i &> hence xj f dg Â» xLfdg. (In fact, from some index n on we ' I n J I
PAGE 105
102 have If fl < 1 => for n > n A we have If I S If  + 1. " n " 2 " n ' n ' which is dgintegrable on I.) Similarly, f x Â•+ fx n uniformly, so from some index n^ on, If xfx < Â— Â»> " n "2 for n>n , If xl < If x + 1 < ff llxl 1, integrable since H is a vector space. By Lebesque, then, /j^xdg Jjfxdg. Finally, since  gx  < g.x, f R are d(gx)integrable, so J f d(gx) + Lfd(gx) by Lebesgue as before. For each n we have x T f dg = Lxf dg = Lf d(gx), ; I n ; I n 'In so on passing to the limit we get xj fdg = Lxfdg = J fd(gx), which is (4.3.1 ). iv) Let (f ) be a uniformly bounded increasing sequence of positive functions from H, and denote f = lim f . Show n n f e H. Let M > f for all n. Then we have n J x f n dg Â£ J I Md  g = MÂ«g(I) for all n. By Lebesgue, f is dgintegrable on I, and Lf dg + Cfdg, hence 'In 'I x /l f n d 8 " xjjfdg. Similarly, Jf^J < f n x Â£ M* x  e L (dg), so fx is dgintegrable on I by Lebesgue, and J f xdg * J fxdg. Also, f is d(gx)integrable since f n  < M and J Md(gx) < Mxg(I), so by Lebesgue again
PAGE 106
103 we have f is d(gx)integrable on I and Lf d(gx) * J I n J fd(gx). Now, (1.3.1) holds for each n, so passing to limits as before, (4.3.1) holds for f as well. Now, let S be the family of sets of the form R = ( (s,t) , (s ', t ') ] () R . Since the rectangles ( (s,t) , (s',t ')] form a semiring 2 ? generating the Borel ofield on R , S is a semiring generating B(R ). Let, then, C be the family of indicators of sets of S. To complete the monotone class argument, we must show that CC H and that C is closed under multiplication, as H then contains all bounded functions p measurable with respect to o(C) = B(R ). C is closed under multiplication, as 1 1 = 1 , and S is R. n R O R_ a semiring, so R C\ R e S => 1 e C. Now we show that (1.3.1) I Â£ K i i H holds for f = 1 , R e S. We have H xj^pdg = x/l Rni dg = x(g(ROl)) (Again g() refers to the measure dg.) Since x(g(ROl)) Â£ xg (ROD S xÂ»g(I) < Â», 1_ is dgintegrable and d(gx)integrable. Also, JjXl R dg = Jx1 Rn;[ dg = x(g(RPlI)), and f I 1 R d(gx) = Jl Rni d(gx) = (gx)(RplI) = xtglRfil)), hence x/j1 R dg = JjX^dg = / 1 d(gx), which is (4.3.1), so C H. This completes the proof for f bounded, g(I) < Â°Â°. Assume now, g(I) < *, f dgintegrable on I (not necessarily bounded). There exists a sequence (f ) of bounded functions n
PAGE 107
ion converging to f a.s. and in L 1 (dg) on I, with (4.3.1) satisfied for each n. In the first integral, we have ft dg Â»Â• Lfdg, hence x/j^dg x/jfdg. Also, I/ I xf n dg JjXfdgl = / I x(f n f)dg I x lJ I l f n ~ f l d ls * Â°; hence x I f n dg Â»Â• xjjfdg. Finally, f is d(gx)integrable [5, Theorem 4, p. 172], and we have f f d(gx) / fd(gx)  = l/^^DdCgx)! < /jIVflcKlellxl) xJ I f n f d(g ) as n Â» Â«. Then J^dKgx) + Jjfd(gx). Since (4.3.1) holds for each n, we have it for f as well by passing to the limit. 2 Finally, let I e B(R + ) , f dgintegrable on I. There exists a sequence (I ) of sets from B(R 2 ) with g(I ) < Â», and I t I. n + ' ' n n Then f^ >Â• f1 a.s. and in L (dg) and L 1 (d( g x J ) ) by Lebesgue n (since ri x [ < [f.1  E L 1 (dg) and L 1 (d  g x[ ) ) ; (4.3.1) is n satisfied for each I , so we pass to limits exactly as above. This completes the proof of (4.3.1). The proof of (4.3.2) is completely analogous (since  < xzg). I We conclude this section with the theorem establishing the correspondence between stochastic measures (Pmeasures) with finite variation and processes of integrable variation for the case of realvalued processes and measures. Although this is a special case of the more general result we will establish later, it cannot be deduced from that, since our proof for the vectorvalued case will make use of the realvalued result. Theorem 4.1.4 . There is a onetoone correspondence X +Â» \i between 2 raw processes X: R + xti * R with raw integrable variation x and
PAGE 108
105 stochastic measures u with finite variation p v , given by the A ' X ' equality (H.l.1) u Y (A) = E(J A dX ) for A bounded, measurable. X R 2 z z + Remark . We shall later prove the equality p. . = p  for X with X  X values in a Banach space, from which the equality follows for realvalued X as a special case. Proof. We remark first that the correspondence is onetoone in the sense that we identify processes that differ only on an evanescent set . 1) Let X: R + xQ > R be a raw process with raw integrable variation ]x. For any bounded measurable process A, a < M, the map w * J A (w)dX (w) is in L 1 (P). In fact, we have R ^ z z + J A (w)dX (w) < / A (w)dx (w) < Ml X I (w), and by assumption R R^ Z z + + X e L (P). Then, for any M e M = B(R 2 )xF, E(f Â„1 w dX ) exists. + ; Â„2 M z R + Set, then, u Y (M) = E(f 1 u dX ). Then, for step functions X J 2 M z rv + n n B = I a 1 , M e M, a e R, we have y (B) = JBdy = I a.y(M ) = 11 i * * x ' i=i x l n = E(J ( I a 1 ) dX ) = E(f B dX ). Now, suppose A is bounded, R* i=1 l n i z z R^ z z ) dX )) z z
PAGE 109
106 measurable. Let A be a sequence of step functions, converging uniformly to A except on an evanescent set. For each n, we have u (A ) = E(J Â„A n dX ). Since A is bounded, EC f A dX ) exists, as WÂ€ X 2 z z J z z + have shown. Moreover, E(J 2 A^dX z ) E(J 2 A z dX z ) = E(J 2 (A n A) z dX z ) R . R R S E(J A n A dx ) R z z S (sup A^A z l)x oo zeR 2 + as n Â•* Â« by uniform convergence. Thus E( (A n dX ) * E((a dX ). For ' z z ; z z each n, the double integral is equal to u Y (A n ), and u (A n ) > u v (A) (since y has finite variation, which we shall prove in a minute), hence we have equality in passing to the limit, i.e., M X (A) E( R2 A z dX z ). Now, we must show that (i) y is a stochastic measure, and (ii) u x has finite variation. The first is easy: if M e M is evanescent, then M(w) = for almost all w, hence { 1 M (w) dX (w) = Pa.s., so E(f(1J dX ) 2 M z z **' i M z Z' 0,
PAGE 110
107 i.e., n x (M) = 0. As for (ii), let M e M, (M.) i = 1,2,...,n disjoint n sets form M, with (JM.C M. We have then E y (M )  1 i1 X l l ! 1 ,E(/ H 2 (, M 1 ) . d vi * .= * = e( .V r 2 (1 m >, 11 a + 1 11 Kl 1=1 R 1 + + + E M ?' 1 n m ) 7 d l x l ) since all sections are disjoint = E(f (1 ) dlxl ^ ^ M i z ; R 2 M z I 'z + + The last integral is independent of the family (M ), so by taking supremum we get u x (M) < E(J 2 (1 M ) z d l X U ) " E( I X U < "Â» so ^ x haS R + finite variation (in particular y  is bounded by E(x )). Uniqueness of y is evident: this completes one half of the correspondence. Next we prove the converse. 2) Let p be a stochastic measure with finite variation y. Assume, first, y>0 (then u = y). We will associate an increasing process X (increasing in both senses) satisfying (4.14.1); then the final result is an easy consequence of the decomposition of measures with finite variation. o For each bounded r.v. Y and u e R , consider the raw process Y (w) = Y(w)Â«I ,,,(z). The map A defined by A (Y) = y(Y ) = " L U , U J U u y C Y*IjQ u j) is a bounded, positive measure on (ft.F). In fact: for B e F, A (B) = M CI 1 r ) = y(Bx[0,u]) < y(Qx[0,u] ) . Also, A is PU B LU,UJ u absolutely continuous; if P(B) = 0, then A (B) = y(Bx[0,u]) = since y is a stochastic measure. Then A has a density a with respect to u u ^ P.
PAGE 111
108 Now, if u a.s. In fact, for B e F, E(1_A D (a)) = E(1 a , fc ,) n D n D S t E(1 B a s't ) " ^Vst' 5 + E0 B a st ) " Vt' (B) " Vt (B) " A sc' (B) + X (B) = y(Bx[(0,0),(s',t')]) u(Bx[(0,0),(s',t)]) y(Bx[(0,0),(s,t')]) + p(Bx[(0,0),(s,t)]) = y (Bx( (s, t) , (s ', t ') ] ) > 0. Since the fou" functions that make up A D (a) are Fmeasurable, we have A R (a) ^ a.s. (i.e., for w t N) for R with rational coordinates. Now, we show that for w t N (see note preceding page), we have . 1 1 2 A n (a (w)) Z for all R = ( (s,t) , (s ',t ') ] C R . Let w t N and suppose * Outside an evanescent set: for uÂ£v, u,v rationals, a Â£ a a.s., u v so for each pair u,v there is a negligible set N outside of which a (w) Â£ a (w). We put these together into a common u v negligible set N outside of which a (w) 2 a (w) for any u,v 2 u v rationals in R .
PAGE 112
109 A (a (w) ) < t < 0. We can find, since a^(w) is increasing (on the rationals) on this set, nationals p If A ((s,t),(s',t')] (a1(w)) < e, then we have A.. . , , ,.,(a(w)) = a , ,(w) a , (w) ex ,(w) + a (w) ( (PÂ»q) .(P ,q )J P q p q pq pq Â£ a ,. ,(w) a. (w) a ,(w) + a (w) st p q pq st (by definition of a ) Vt' (w) (a st (w)+ ^ +a st (w) ( (s,t) , (s ,t )] 2 2 2 < 0, i.e., A., . , , , N ,(a(w)) < 0, a contradiction. Then a' is ( (P.q) . (P .q ) J increasing in both senses for w i N. 2 1 1 Now, for each z e R , we have a.s. a Â£ a Â£ a . Also, the + z z z+ map z *Â• X (Y) is right continuous. In fact, lim X (Y) = z u u+z lim p(Y(w) Â• I ,(v)) = u(Y(w) Â• I_ ,(v)). Then uiz [ Â°' U] C Â°' z] X = a is also a density of X^ with respect to P. In fact, on one z z+ t hand, since a < a , for B e F, X (B) = E(1_a ) ^ E(1 D a 1 ) = z z + z B z B z+
PAGE 113
110 E(1 X ). On the other hand, E(1 X ) = E(1 D a 1 ) = E(1 Him a 1 )) o z a z ti z+ 3 u u+z = lim (E(1 a )) (by monotone convergence) Â£ lim (E(1 a )) = , l^ H u B u U+2 U+z lim A (B) = X (B) by right continuity. Putting the two together, u + z A (B) = E(1 X ) for B e F, i.e., X is a density for X . z B z z J z Then for processes of the form A (w) = Y(w) Â• I r ,(z), z [0,u] u(A) = E(J A dX ), since E(J J dX ) = E(f _Y(w) Â• I. ,(z)dX ) a d z z r 2 z z d 2 [0,UJ Z = E(Y(W) '^[O.u^V = EU(W) ' X u ) " V Y) * ^ (Y ' I [0,u] ) = Hlil+ Moreover, a = lim a , so a is also (same proof) z+ z z+ Vi u+z u rational incrementally increasing for w t N; hence X = a is an intestable z z+ increasing process: more precisely, x. . (w) = X. n <,( w ) + X (s,0) (w) + X (0 t) (w) + X (s t) (w) " 4X (s t) (w) f0r W * N ' SO outside an evanescent set x < 4u(R + xft) (we shall use this later). To prove the equality (4.4.1) for A bounded, we use a monotone classes argument. Let H be the set of all bounded, Mmeasurable processes A (w) for z which (4.4.1) holds. H is clearly a vector space. Also, H contains the constants: If A = c constant, then A = lim c Â• I r , . (z), n+0B [(0,0) ,(n,n)] and we have shown (4.4.1) for these processes already. We get the result in the limit by monotone convergence. More precisely, denoting A (w) = c Â• I,,. .. . si(z), we have, for w outside a negligible z L (0,0) , (n,n)j s B set, sup] A (w)dX (w) = supX (w) < *>, so by monotone convergence ' z z z (n,n) Â° K
PAGE 114
11 1 we have J A (w)dX (w) * J .A (w)dX (w) Pa.s. Then, the maps >2 z z ' n 2 z z K R + R + w + J .A (w)dX (w) increase to w * [ A (w)dX (w) Pa.s. and the r; z z V z latter is integrable as we saw above, so by monotone convergence we have that E(f A n dX ) + E(f A dX ). Since u (A n ) u(A) and (4.4.1) R Z Z R Z z + + holds for each n, (4.4.1) holds in the limit as well. Also, H is closed under uniform convergence: let A be bounded, measurable, A a sequence from H with A * A uniformly. By Lebesgue y(A ) * y(A). Also, for almost all w, X(s) is a bounded positive measure, so J S (w)dX (w) * I A (w)dX (w) Pa.s. (by Lebesgue again) so by R R + + Lebesgue the map w Â»  _A (w)dX (w) is Pintegrable (since each R 2 Z + A n z H) and E({ _A n dX ) ^ E(f J dX ). Since (4.4.1) holds for each R z z R z z n, it holds in the limit as well. Finally, let A be uniformly bounded, A" +A, A e H for all n. As above, using a double application of the monotone convergence theorem this time, we have E( [ A' dX ) R 2 Z Z + E(J A dX ) , and p(A ) Â» y(A), and we conclude as above. R ^ z z + To complete the monotone class argument, let C be the class of processes of the form Y(w) Â• I ,(z). We already know C Â— H; it is easy to see (taking Y indicators of sets of F) that o(C) M. Finally, C is closed under multiplication; in fact, U\w) Â• I [0iU] (z))(Y 2 (w) Â• I [QtV] (z)) = (Y 1 (w)Y 2 (w)) Â• I [0>uv] (z) e C, and the monotone class argument is done. This completes the
PAGE 115
112 proof for \i Â£ 0; if p has finite variation, we write y = y u + + and associate X with y and X with y . We have, then, for A bounded, measurable: y(A) = y (A) y (A) = E( f A dX + ) R 2 Z Z ' E(J .A dX _ ) = E(J A dX + f _A dX~) = E({ A d(X + X _ )). Setti ' . ? 7. 7. J ? 7 7 J ? 7 7 J P 7 7 7 ng X = X + X", y(A) = E(J A dX ), and x = X + X~ < X f  + x", R ^ z z + so X has integrable variation, and the theorem is proved. B Remark . In Meyer [12] a version of this theorem (without proof) is 2 given for Pmeasures and random measures on R xft. 4.2 Measures Associated With VectorValued Stochastic Functions In this section we shall show that, starting with a stochastic function, we can associate a Pmeasure, with finite variation if the function has integrable variation. Our first theorem is for measurable processes with integrable variation. Theorem 4.2.1 . Let E be a Banach space, X an Evalued, raw, right 2 continuous process such that X is integrable for every z e R , with raw integrable variation l x l(s,t) = l x (o,o)S +V8r co i 8] (x (.,o) ) + Var [o,t] (x (o,.) ) + Var [(0,0),(s,t)] (X) 2 There is a stochastic measure (Pmeasure) y : B(R )xF * E with finite variation satisfying the following.
PAGE 116
113 If $ is any scalarvalued measurable process, we have * e L (y v ) if and only if E(( Ai> Idlxl ) < Â«. In this case, n + E(( dX ) is defined, 'r 2 Z z (1.2.1) y (*) E(f $ dX ), and X J R 2 z z CI2.2) y x (Â») = E(J 2 * 2 dx z ), R i.e., y x  = p x . Proof . For M e M, the integral E([ Â„1 tJ dX ) is defined; to see V M Z + this, we use a monotone classes argument. More precisely, let H = {M e M: E(J Â„1 w dX ) is defined}. We will show that H is an V M z + algebra, is closed under monotone convergence, and contains a semiring generating M; we then conclude from the monotone class theorem that H Z5 M, hence H = M. H is an algebra . Let A,B e H; show AVJB, APlB, A c c H. First of all Â» / o 1 A (w ) dX Cw) exists Pa.s. as well as J 1 (w)dX (w). r; A z R 2 B z Then 1.(w) Â• 1 R (w) is dX (w)integrable almost surely, i.e., J 2 1 A (w) Â• 1g(w)dX (w) = J 1 n (w)dX (w) exists Pa.s. Moreover, R + L Z R + I/ 2 1 AnB (w)dX z (w) l ' /Â„ 2 1 AOB (w)d l X !z (w) ' / 2 1 A (w)dx z (w); hence R , R R + + + w * J o 1 A^ Q Cw)dX (w) is Pintegrable, i.e., E(f J s ^ n dX ) exists, ' 2 Add z J 2 AflB z
PAGE 117
m so JIObs: H, We get A (J B e H by writing 1 =1+11 W ' 6 AUB A B AHB and A C by 1 = 1 i (E(fldX ) = E(X )). ,0 A ' z " A H is closed under monotone convergence. Let A be a sequence of sets Â— . Â— Â— Â— n from H with A increasing to A. For almost all w, f Â„1 . (w)dX (w) * a 2 A R n J 2 1 A (w)dX z (w) by Lebesgue (since 1 < 1 e L 1 (x(w))). For each n, R + n the map w * Jl (w)dX (w) is bounded by ^^(w) e L 1 (P) , so these n converge to w Jl (w)dX (w) a.s. and in L (P). In particular, E(Jl (w)dX (w)) exists. The proof for decreasing sequences is the same. H contains a semiring generating M . Let S be the semiring of halfopen rectangles in the plane from before, and denote P = sDn. Then U = {AxF, A e P, F e j} is a semiring generating M. For a set B e U, not only is E([l dX ) defined, but we can compute it 1 B z explicitly. There are four types of sets in P (cf. Theorem 3.2.2): 1) A = ((s,t),(s',t')]: then E(/l, ^.dX ) = E(1_fl,dX ) " AxF z F J A z = EM Â• (A.(X))). F A 2) A = (s,s'] x [0,t']: then we get E( [l . dX ) = E(1 (X , X, ,..)). F (s ,t ) (s,t ) 3) A = [O.s'l x (t.t']: we get E(Jl dX ) = E(1Â„(X , X. . .)). F (s ,t ) (s ,t)
PAGE 118
115 1) A = [0,s'] x [0,t ']: we get E((l dX ) = E(1 X, , ,J. J AxF z F (s t ) ( Note : See Theorem 3.2.2 and preceding example for computations of the measure dX(w) on these rectangles.) By the Monotone Class Theorem, H contains o(U) = M, so E( fl dX ) J M z is defined for all M e M. Set p v (M) = E(( _1Â„dX ). Then y v : M > E X ' 2 M Z X K + is a oadditive stochastic measure: p is evidently additive. If M + , then for each w, ]1 (w)dX (w) + by aadditivity of the n ' M z n integral. Then E(/l dX ) * by Lebesgue; hence p Y is aadditive. n Z X Also, if M is evanescent, then M(w) is empty Pa.s. => fl (w)dX (w) = ! ' M z a.s. => u x (M) = E({l M dX z ) = 0. Now, y>; also satisfies Ju x (M)  < U X (M), since  u (M)  = E(/l M dX z ) < E(/l M dX z ) < E(/l M dx[ z ) = M x j CM) ( U  X  is the measure associated with x by Theorem 4.1.4); hence p has finite variation u x l U x  (since the variation is the smallest positive measure bounding the norm). We shall prove this is an equality. Each X , being measurable, is almostseparably valued, so we can find a common negligible set N rt outside of which X is separately z valued for z rational. By right continuity, X = lim X , z u u+z u rational so for w t N , X takes on values in a separable subspace EC E. Let Z C E' be a separable subspace norming for E n . Since u x  Â£ u I x I which is finite, we have p << u Â• . . By the extended RadonNikodym Theorem (Theorem 1 .5.8) , there exists a stochastic function jf: R 2 xJ2 > Z' ( = L(R,Z')) having the following properties:
PAGE 119
116 1 ') H is y, .measurable and h < 1. In fact, Theorem 1.5.8 says that Jh is y , , integrable and that for \\i e L (u v )  X  X we have Ji^dy x  = J H^dp i x  Â• Taking tj; = 1 A , A e M, we obtain u x (A) = /h Â• 1 A dy x . < W X (A); hence h < 1 on A, so h Â£ 1 except on a y, ,negligible set (on which we l x ! modify H appropriately, say by setting H = 0). 2') is y,  integrable for every z e Z, and we have = J dy, , for every M e M. In fact, taking f ! 1 in Theorem 1.5.8 (2), we get y, , integrable for all z. Also, for z e Z and M e M, we have, taking f = 1 , = Jdy x , i.e., = /l M dy  x  (since = 1 ) = f dy, ,. M M ^M  X I 3') y (M) = J H dy I  for M e M. We showed this in proving (T). Now, taking M = [0,u] x A, A e F, in (2'), we deduce that (4.2.3) EM ) = E(1 L ,dx ) A u A J [ 0,u] w ' ' w In fact, on the left hand side of (2'), we get = = = = ; R 2 M w A J [0,u] w A u J A u +  A dP = E(1 A ), which Is the left hand side of (4.2.3). As
PAGE 120
117 for the right hand side, Ldy , v , = E([l dlxl ) (by Tneorem J M [X ' M w w 4.1. iÂ», since  < Hz S \z\) = E(1 A / [0>u] y d X \J , which is the right hand side of (4.2.3), thus proving the equality. Now, since (4.2.3) holds for all A e F, there is a Pnegligible set (depending on u and z) N(u,z) C Q outside of which = J r , dlxl . Since both sides of this equation are U ; [0,u] v ' ' v right continuous, there is a negligible set N9x) = [) N(u,z) u rational outside of which = \ rn , dlxl for all u e R 2 . u ; [0,u]v li v + Let S be a countable dense subset of Z and set N = (J N(z); zeS N is negligible. Also, since x is integrable, there is a third negligible set N outside of which Ix^U) < Â». Let, now, w t NgU^U^ be fixed. The function X # (w) = X(w) is an E valued function defined on R + , having bounded variation ^(w)^ = [ X  _ ( w) . Then (Theorem 3.1.1) it determines a Stieltjes measure y . on X ( w) B(R ) with finite variation u , J. By Theorem 3.2.2, we have + i x(w) ' \v vf \ = V\ v , \iÂ« Then by Theorem 1.5.7, there exists a function mw; mwj 2 G : R. > Z such that 1M) i G wÂ« 1 ^ixwr a e 2") is \i, . . integrable for every z e Z and W  A(,W j  = f M d Mx(w) for M e B(R 2 ) . In fact, taking f = 1 in 1.5.7(2), we obtain (as for (2')): M
PAGE 121
113 = K 1 M' Z>d ^X(w) = / 1 M dy x(w) ** /m ^x(w)Taking M = [0,u], we obtain = , and X(w) u ^ dy x(w) = /[0,u] . d l X l. (w); henCe <\^>*> = Â•MO ul d l X l Â• ^ * Puttin 8 this together with what we had 2 earlier for , we now have, for Z e S, u t R , J[Q,u] . d l X !. (w) = <^ u (w),z> J [0U] j d ]x > (w). 1 _ 2 There is then a m^,. , negligible set N (w,z) C R + outside of which we have z> = . In fact, the two integrals above form measures on B(R + ). By taking differences, we have, for ud l x l (w) = !r .,dx (w), and these rectangles, along with those [0,u] generate B(R ); hence = + w y x(w)] a.e. Now, the set N (w) = [) N (w,z) is y . , ..negligible, and for 1 zeS l X(w) l u t N (w) we have = for all z z S; hence for all u w z e Z since S is dense in Z. Since Z is norming, we have H (w) = G (u) for u t N 1 (w). u w Let A = {(u,w): H (w) I < 1 } . A is then y measurable (in fact, A = A U N with A Q e M, N y x , negligible. Then N is y x negligible so A is y measurable). For each w, consider the section A(w) = {uH u (w) < 1}; since for w t N JjN.ljN we have G wl = l p x(w)" a e ' and H u (w) = G w (u) y x(w)" a ' e " we deduce that A(w) = {u: H u (w) < 1} = u: G w (u) < l} (a.e.) is V x(w) negligible. Then U X (A) = E(J g 1 A(w) (u)d x  (w) ) = since R
PAGE 122
119 J 2 1 A(w) (u)d l X 'u (w) = Â° P_a S * Then l H I = 1li x ia.e.; hence by (3') we have w x (M) = J M H Jclu i x = / M 1<*M X  ^ X  (M) for M E M > i.e.,  vi  = u i i The remainder of the theorem ($ e L (u v ) iff E(l Â„ Jdlxl ) < Â«) will be proved in the next theorem in more n + generality. I Proposition 4.2.2 . Let E,F be Banach spaces and V: R xQ Â» L(E,F) p be a process with V integrable for every z e R and with raw integrable variation v. If X is an Evalued measurable process, then x e LÂ„(y.,) iff X is u. . almost separably valued and E V X E (J 2 l X z l d l V l z ) < "* In this case Â» E(J X dV ) is defined, and R + R + UÂ„(X) = E(fx dV ). V J Z z Proof. One way is easy: if X e l1,(h ), i.e., X e L^(y,Â„,), then X is t, v b V M v almost separably valued (being measurable) and E({ x dv ) R + < Â». In fact, if X e Lg(u v ), then x 1 = U i v i C x  ) = l e Cm v } E U ?l x l d l v l ) b y Theorem 4.1.1 (which holds for positive measurable R^ z z + processes as well by monotone convergence), and this is finite by assumption. For the other implication, let X be an Evalued, measurable process, ui .almost separably valued and satisfying the condition E(J ol x l d l v l ) < "Â• Let (X n ) be a sequence of u ,Â„, measurable step ' _2 z * ' z Mm + processes such that x n > X u, ,a.e. and [ x n J < x [f for every n. Let
PAGE 123
120 A be a \i, , negligible set outside of which X is separably valued and X * X pointwise; then u.,.i(A) = E( [ J.dlvl ) = => I v I J 2 A ' ' z J 1 (w)d  V  (w) = Pa.s. Denote the exceptional set by N. For IT A z + w i N, the section 1.(w) is d  V # (w)negligible, so X (w) X (w) dv # (w)almost everywhere, and x"(w)  < x (w) J. Now, since E(JX z dv ) < Â», there is a Pnegligible set N 1 e F such that for w t N 1 , / x (w)dv (w) < Â», i.e., X(w)  is dv (w)integrable. Then for w t N {J N we have x n (w) S X(w) e L 1 (dv(w)) and x n (w) [ > X(w) dv .(w)a.e. so by Lebesgue / x n (w)dv (w) + / x (w)dV (w) for R^ z z R* z z w ef N^JN 1 and in particular J X n (w)dV (w) Â»Â• f X (w)dV (w) ' 2 z z J 7 7. v. R 2 Z R 2 Z for w Â«f N VJN 1 . Repeating the procedure, since the map w * J x (w)dv (w) R 2 " Z is Pintegrable, JX^(w) d  V  (w) < /X (w)dV (w) Pa.s. and /xj(w)dv z (w) + /X z (w)dv z (w) Pa.s., we can apply Lebesgue again and deduce that we also have convergence in L (P) j in particular, E(J x n dv ) >Â• E(J X dv ). Moreover, for each n R^ Z z R^ Z z we have E(l/x^dV z ) < E( JxÂ£d Vj ) < E(JX z dV z ) < Â•; hence JX z (w)dV z (w) e L (P) for each n. ( Note : We must show this map is measurable; this will come out of a later computation.) We already showed that Jx n (w)dV (w) > fx (w)dV (w) P a.s. so by Lebesgue the z z J 7. 7.
PAGE 124
121 limit is Pintegrable and we have E( [x n dV ) * E( fx dV ). (in ' z z ' z z particular E(/x dV ) is defined). Next, we show that x e L 1 (u ) : z z g ^Y for each n, M v , < !lx n  ) = E(JxJjdv z ) < E(JX z dv z ) < , so by Fatou we have M .. (lira infx n fl) < lira inf M  v (jx n [) < , in particular lim inf Jx n  is M , y , integrable. But [x = limx n I a.e. so X is y , y , integrable. Also, x = lim X n is y measurable, II n V so X e L E (u v ). Moreover, y v (X n ) > u (X) by Lebesgue again, since x n B S X e L 1 (u v ). Finally, we show that, for each n, we have u (X n ) = E( fx n dV ) V ; z z ' so we get the desired equality by passing to limits. Being a step k process, we can write x n = E 1 X. , M. e M, X, e E. Then we have 11 M i i i V^1 = V E1 M V = Zx iW = Lx i (E( / 1 M dV z )) (by Theorem ^2.1) i i = SE(xJl M dV z ) = EE(/1 x dV ) (by Theorem K.I .3) = E(/(E1 x )dV ) i i M. i z = E(Jx"dV ) (and in particular the map [x n (w)dV (w) is Pz z J 2 i measurable). Letting n * Â», we obtain uÂ„(X) = E((x dV ), and the V ' z z theorem is completely proved. I Remarks . 1) By taking E = R in the statement, we have the following: If X is any scalarvalued measurable process, then X e L 1 (u ) iff
PAGE 125
122 E (J I x l z d l v l z ) < "Â• ( x is automatically separably valued.) Then E( J' X z dV z ) iS defined ' and M y (X) = E(Jx dV ). Finally, equality (4.1.2) is proved the same way as (4.1.1), by taking step processes and passing to limits. 2) The correspondence is not onetoone: as we shall see in the next section, a stochastic measure with values in L(E,F) is generated by a stochastic function(not necessarily measurable) with values in a subspace of L(E,F"). We can also generate stochastic measures from stochastic functions (not necessarily measurable) with raw integrable variation, as the next theorem shows. Theorem 4.2.3 . Let E,F be two Banach spaces and Z C F' a subspace 2 normmg for F. Let B:R + xQ Â• L(E.F) be a rightcontinuous stochastic function satisfying the following conditions: i) B has raw integrable variation b. ii) For every x e E and z e Z, is a realvalued process (measurable!) with raw integrable variation . Then there exists a stochastic measure m: M * L(E,Z') with finite variation m satisfying the following conditions: 1) If X is an Evalued process and if X is u ,_ , integrable, then 1 l B l X e L (m) , the integral E() is defined for every fc J d u u J H + z e Z, = E(<[x dB ,z>), j ii ii
PAGE 126
123 and m C X1> S E(/xJdB u ). 2) If, in addition, Bx is separably valued for every x e E and if X is y, ,integrable, then the integral E([ x dB ) is defined, and B J R 2 u u + m(X) = E([ X dB ). V U U + 3) The measure m has values in L(E,F) in each of the following cases: a) F = Z\ b) For every x e E and v e FT, the convex equilibrated (balanced) cover of the set (b (w)x: w e flj is v relatively o(F,Z)compact in F. c) For every x e E and v e R , the function B x is F+ v measurable and almost separably valued; in particular, this is the case if F is separable. Proof . Let u, be the measure generated by 3 via Theorem 4.1.4. For every x e E and z e Z the variation of the process satisfies  < b xz (cf. proof of Prop. 4.1.3). Let m be the stochastic measure generated by : m Y 7 (M) = E( / 5 1 M d) for M e M. The mapping (x,z) Â•> m (M) A Â» Â» D^ x , z + is linear in each argument: in fact, m (M) = X + X 7 1 2 ,Z E(J 2 1 M d) = E(Jl M d) = E( Jl M d( +) )
PAGE 127
121 E (/ l M d+/l M d) = E(Jl M d)+E(/l M d) = m (M) + m (M). The computation for z is completely analogous. x i Â» " x i z Also, we have m (M)  = e(J 1 M d)J < EC if 1Â„d) < x , z. ' Â£ nv ^ Â£ n V R + R + B(J 2 1 M dl) < E(J 2 1 M xzdB v ) = x.2.E(Jl M d8 ) R + R + = 1 X B Z 1U B (M) , so m Xj2 (M) < xzui B i(M). Then for given M e M, the map (x,z) m (M) is continuous, bilinear. Then there is a continuous linear map m(M) e L(E,Z') satisfying = m x (M) = E(/ 2 1 M d ) , (More precisely, any continuous, bilinear function f(x,z): ExZ Â•Â» R is continuous and linear in each component. Then the map x Â•Â» f(x,Â«) is a continuous linear map from E into Z'. In our situation, we have m(M)x = m x ^(M), so = m (M), i.e., for M e M, we have m(M) e L(E,Z').) We also have m(M) I < p. ,(M), since B m(M) = sup m(M)x = sup m (M)  = sup ( sup m (M)) < xS1 x<1 x Â«* fx<1 Jz<1 x ' z sup ( sup ]x zp . , (M) ) = y.(M). This gives us a map x<1 Jz<1 l B l l B l m: M L(E.Z'). We now verify that m is oadditive and has finite variation m (in particular m < \i. .): First of all, m is additive . Let M,N e M be disjoint; we show m(MlJN) = m(M) + m(N) , i.e., that m(M\JN)x = m(M)x + m(N)x for all
PAGE 128
125 x e E. This amounts to showing that = for all x e E, z e Z. Now, = E(f 1 d) R 2 MUN v E( f ? (l M +1 N )d) " E( / 2 1 M d< V' 2> + / 2 1 N d< V' Z> E( I 2 1 M d) + E( J" 2 1 N d) = z> + = R + R + ' , which shows that m is additivie. If, now, A n +0, m(A n ) since Jm( A n >  < ji. .(A ) and the latter is oadditive. Then m is oadditive as well. As for the variation, \i. , is a bounded, positive measure B satisfying m < u , B ; hence m < u i B since the variation is the smallest positive measure dominating the norm. In particular, m has finite variation. Now we prove assertions (1)(3). Ad (1 ) : Let X be an Evalued process. From the inequality 4(u B ), then x E 4< 4"B Cauchy in L^(m); hence X e Lg(m) since X is Mmeasurable, and m(x) < U  B (xJ) = E(J 2 xJ v dB v ), which is the second part of (1). The first part of (1) is satisfied for any Mmeasurable step process X = I 1 x , M e M disjoint, x. e E. In fact, for 11 i 1 z e Z, we have = < I m(M )x ,z> 11 M i i 1=1 i i m < Uj B  it follows that if x e ^(U B ), then X z L^(m) (since any sequence of step functions Cauchy in ll\i, , is then also
PAGE 129
126 n n t n = I = 1 E(J 1 d<8 x z> ) = EEÂ« 1 dBx.,jÂ»( 1=1 1=1 fT M i v 1 i=1 R 2 M i v x + + n Prop. 4.1.3) = E( E ) = E() = i=i Â• v 1 J M v l by r t n E( n. l v ; . , M, l v i=1 i E(). Now, let X e L^y^,) and let X n be a sequence of measurable step functions such that x" X u. .a.e. and x n  < j X  everywhere. Let A be a y.. negligible set outside of which X is separably valued, / X^ jd  B  v < Â» (more precisely, for (w,u) i A, / [0 , u ]l X v (w) ! d l B lv (w) < Â°Â° ; 3ince E( / 2 l X v I d l B ly ) < ". R + / 2 l X v (w) i d l B l v (w) < " p a s : hence J [0(U] X y (v) dB y (w) < Â« + y I B I ~ a * e * since i,: is a Pmeasure), and X n > X. There then exists a Pnegligible set N t F such that for w i N, the section A(w) is dB (w)negligible (in fact, E(fl , .dlsl (w) ) = y, .(A) = => J A (w; ' ' v 3 J 2 1 A(w) d ' B l. (w) = Â° p ~ a Â« s )Â» so for w ^ N we have d  B  (w)a.e.: R + i) Xjw) is separably valued ii) X%) < X.(w) iii) x"(w) * X/w). Since / X (w) d B  (w) < Â» for w t N, X (w) is d3 (w) R v + integrable, and x"(w) X (w) in L.J,(dB (w)), so by Lebesgue
PAGE 130
127 Jx"(w)dB (w) > JX (w)dB (w). Then, by continuity, for w ^ N, z e Z, we have Â•* < f X (w)dB (w),z>; moreover, R 2 v 'R 2 v v + + l * NI*l/x"(w)dB v (w) < z./xJ(w)dB y (w) i ll z l/l x v ( w )I d l B l v ( w )Â« Now, the function w / X (w)dB (w) is PR 2 v v + integrable, so by Lebesgue is Pintegrable and R V V + E(< / ? x"(w)dB (w),z> E() for all z e Z. For each R R V v + + n, = E(<[ _X n dB ,z>) as we saw above. Finally, X is J d v v J H + Mmeasurable by assumption, and flx e L (\i> i)CL (m); hence X e L Â£ (m), and m(X n ) m(X)  < m(x n X) ^ by Lebesgue (since x n  < x, X n X ma.e.), i.e., m(X n ) m(X). Then * for all z e Z. Passing to limits, we obtain = E() for all z e Z, which completes the proof of (1). R v v + Ad (2) : Suppose, now, that Bx is separably valued for every x e E. Let X c L (pi ), let X be step processes convering to X yj.a.e. with [X  < Jx for all n. We shall show first of all that the map w * J X (w)dB (w) is integrable for all n; write R v X = I 1 x , M . e M disjoint, x e E. For each i, Bx. i=1 i l 1 is separably valued. Also, by (ii), is measurable for z e Z. Since Z is norming, Bx is weakly measurable, so Bx , being separably valued, is
PAGE 131
128 strongly measurable, with integrable variation (in fact,  Bx  S Bx  by Prop. 4.1.3). By Theorem 4.2.1, E(f 1 d(Bx.)) exists. 1 ' Â„2 M. l R i n n Then I E(J 1 d(Bx )) = I E(/ (1 x. ) dB ) (by Prop. 4.1.3) 11 R i 11 R^ i * u u + + n n E( I Jl x dB ) = E(J( I 1 x.)dB ) = E(fx"dB ) exists. We proved i=1 M i : u 111 u u in (1) that Jx"(w)dB (w) Â•* Jx (w)dB (w) Pa.s. Moreover, for each n, /x"(w)dB u (w) < Jx"(w)dB u (w) < /X u (w)dB u (w). By assumption, the latter is Pintegrable, so by Lebesgue Jx (w)dB (w) is Pintegrable, and we have E(fx n (w)dB (w)) Â•* E( fx (w)dB (w)). We have J u u ' u u from before that m(X ) Â» m(X). It remains to prove that m(X n ) = E(/x n dB ) for all n. For all z e Z, we have = k = = Km(M.)x.,z> = Im (M ) = 1 = 1 i i l X1 x.,zi *E(Jl M d) = ZE() = EE( z>) = E(Z) = EÂ«/(Z1 M Xi )dB u ,z>) = E() = . ( Note : We can now do this last step since Jx"(w)dB u (w) is Pintegrable; it was not in part (1)!) Both m(X n ) and E(Jx n dB ) are Z 'valued, so this means that m(X n ) = E(J 2 X u dB u^ for a11 n " Pa3sin 8 t0 the limit, we obtain R + m(X) = E(J X dB ) ; in particular, the double integral on the right is R + defined.
PAGE 132
129 Ad (3) = (a) is trivial. (b): Let x e E, v c R . Since the set C = coJB (w)x: weftl + y J ( balanced closed convex hull) is o(F,Z)compact , the natural embedding of C in ZÂ», the algebraic dual of Z, is a(Z*,Z)compact (see Dunford and Schwartz [8]). There is then a family (z.). , of elements of Z such that l id C = ( ){yeZ*:  < l} (any closed convex set is an intersection iel of halfplanes; we can use balls since C is equilibrated). Then we have  < 1 for all i e I, w e R. Let M = [0,u] x A, A e F; we have  \ = E(/l { ^ u]xA d )  = E(1 A ) < E(1 A ) < 1; hence m([0,u]xA)x e F, i.e., m([0,u]xA) t L(E,F). By taking differences, we have m((u,u']xA) e L(E,F), and also finite disjoint unions of such sets. We shall use the monotone class theorem to prove that m(M) e L(E,F) for all M e M. Let M = {M e M: m(M) e L(E,F)}. We show first that is a notone class: Let M e M , M t M. Then m(M )x e FC Z'. We have n n n (M n )x m(M)x < m(M n ) m(M)x by oadditivity of m. Hence m(M n )x + m(M)x in the metric topology of Z'. Since F is closed in Z' for the metric topology, m(M)x e F as well, i.e., m(M) e L(E,F). The proof for M + H is exactly the same, n Now, let C be the algebra generated by sets of the form (u,u']xA, A e F. C consists of finite unions and complements of such sets. We have shown that if M is a finite union of such sets, then mo lm
PAGE 133
130 m(M) e L(E,F). As for complements, m(MÂ°) = m(R + XQ) m(M), and 2 m(R + xft)x = lim m([(0,0), (n,n)]xfl)x e F by closure of F in Z' as before. Thus, if m(M) e L(E,F), then m(M c ) e L(E,F). Then CC M , C is an algebra, so M = o(C)Cf/ by the monotone class theorem, i.e., m takes values in L(E,F). c) Suppose that for x e E, v e R , the function B x is F+ v measurable and almost separably valued (in particular, if F is separable, then B x is separably valued and weakly measurable by (ii); hence B x is Fmeasurable) . Then for every A t F, x e E, the function 1 B x is integrable; in fact, B x is Fmeasurable, by assumption and we have B xj <, \s  Jx e L (P). We also have, as before, = E(/l [0>v]xA d x,z>) = E( ) = = (again, we can move the expectation inside since 1 B x is integrable). Since this holds for all z Â£ Z, we conclude m([0,v]xA)x = E(1.B x) e F. Then A v m([0,v]xA) e L(E,F), and we conclude by the same monotone class argument as in (b). I Remarks . 1) This theorem shows that if B has values in L(E,F), then m B has values in a subspace of L(E,F"). Moreover, we do not have in general m  = Uigi. Later we will establish some conditions sufficient for equality. 2) The correspondence B Â» m is not infective . For an example involving measures associated with functions, see Dinculeanu [6, p. 273].
PAGE 134
131 4.3 VectorValued Stochastic Functions Associated With Measures In this section we consider the converse; starting with a stochastic measure m with finite variation, we will find a stochastic function B with integrable variation such that m is associated with B in the sense of Theorem 4.2.3. The precise result is the following: Theorem 4.3.1 . Let E,F be two Banach spaces and ZC F' a subspace norming for F. Let m: M Â» L(E,F) be a stochastic measure with finite variation m. Then there exists a right continuous stochastic function B: R"~x8 Â» L(E,Z') satis r ying: i) B has raw integrable variation s. ii) For every x e E and z e Z, is a realvalued raw process with integrable variation . Moreover, we having the following: 1) If X is an Evalued measurable process we have X e L (m) E if and only if X e L (p, ,). In this case the integral E() is defined for every z e Z, V u u = E(), and R 2 u u m C X> = E(J 2 xJdB u ), i.e., m = U( 2) If F is separable (or more generally if Bx is separably valued for every x e E), then Bx is measurable for every x e E. If B is separably valued, then B is measurable.
PAGE 135
132 3) We can choose B with values in L(E,F) in each of the following cases: a) F is the dual of a Banach space H and we choose Z = H; hence F = Z'. b) For every x e E, the convex equilibrated cover of the set {Jijixdm: $ simple process, J$dm < 1 } is relatively o(F,Z)compact in F. c) E is separable and F has the RadonNikodym property (we say F c RNP) ; in this case 3 can be chosen such that Bx is measurable and separably valued for every x e E, hence m(X) = E(Jx dB ) for X e l1(id). J u u E d) The range of m is contained in a subspace G L(E,F) having the RNP: in this case B can be chosen measurable, with separable range contained in G; hence m(<{0 = E(J<{> dB ) for * e L (m). ; u u 4) If p is a lifting of P, we can choose B uniquely up to an evanescent set, such that p[B ] = B for every v e R (see Definition v v + 1.5.5(b)). Proof . Let V be the integrable increasing raw process associated with  m  via Theorem Jl.1 .4: I ml (M) = E(f 1 M dV ) for M e M. II ; D 2 M u
PAGE 136
133 Denote the rectangle [0,z] by R ; for z e R 2 set m Z (A) = m(R xA) z + z for A z F. We verify that m : F Â» L(E,F) is a oad;Jitive measure with finite variation m , and that m Z is absolutely Pcontinuous : i) m is oadditive : First of all, mz is additive. Let A,B z F, disjoint. Then R xA and R xB are disjoint, so we have z z J . m Z (A(JB) = ai"(R x(AUB)) = m((R xA) [J (R xB)) = m(R xA) + m(R xB) z z z z z = m (A) + m (B). Now, let (A ) z F, A +0. Then (R xA )+(R x 0) n n z n z = 0, so lim m"(A ) = lim m(R xA ) = m(0) = 0, so m Z is indeed n z n n n oadditive. ii) m has finite variation : we show in particular that m  < m z , where m z (A) = m(R xA). Let A z F, and let (A.), z i n i = 1 n be disjoint sets from F with \JA C A. We have, since i = 1 n n n n R x((JA ) = M(RxA)CRxA, E h Z (A.) = I m(R xA . ) j < z i=i 1 fii z l z i=r l ii z i n E m(R xA. ) <  m  ( R xA) = Iml (A). Taking supremum, we obtain i1 z 1 z m Z (A) < m Z (An iii) m Z << P : In fact, we have m Z (A)  = m(R xA)  ^ m(R xA) = E(J 1 A dV ) = E(1.V ); hence m Z << P, so z '2 R xA u A z ' ' ' R z + z m << P as well. Applying the Extended Radon Nikodym Theorem (1.5.8), we get, for 2 Â° each z z R , a function B : Q + L(E,Z') satisfying: 1) B z  is Pintegrable, and for ty z L (m Z ), we have f i z, r Â° J4idm  = J B 
PAGE 137
131 2) is Pintegrable for all f E ll(\m Z \) and zÂ„ e Z, z u E ' ' and = [dP. 1 ' z 3) If p is a lifting of L (P), we can choose (B ) uniquely Pa.s. such that p[B ] = B , i.e., for all A e F, x e E, z e Z. z z * we have 1 A e LÂ°(P) , and p ( 1 ft ) = 1 . If, in addition, there exists a>0 such that m Z  < aP, then we can choose Â° Â° Â° Â° oo B uniquely everywhere such that p(B ) B . i.e., <3 x.z > e L (P) z z z z for all x e E, z Q e Z, and p() = for all x,z . 4) If one of the conditions in 3(a) or 3(b) is satisfied, then B takes values in L(E,F). z ' ' Now, in particular, taking i/ 1 , A e F In (1 ) we obtain 1 ') m Z  (A) = E(1 B ) for A e F. Also, taking first f Â» x, x e E, and then f = x1 , A c F, we get 2') is integrable for x e E, z n e Z, and z = = /dP = E( 1 A ) , for A e F, x e E, z e Z. (Notice also that from (1'), if B is bounded, the condition m Z  < aP in (3) is satisfied.) From (1') and the inequality m Z  < m z we obtain EdjBj) = m Z (A) < m Z (A) = m (R^A) = E(/l R xA dV u ) = E(1 V ), z o i.e., E(1 A B ) S E ^ 1 A V Z ) for a11 A c F; hence B  S V Pa.s. Let z = (s,t), z = (s',t') be points in R , z < z'. Denote by D , the set R , \ R , and by R , the rectangle ( (s,t), (s',t ')]. zz z z z z
PAGE 138
135 We have m(D xA) = m((R , \ R ) X A) = m((R # xA) \ (R xA)) = zz z z z z m(R ,xA) m(R xA) = m (A) m Z (A). Likewise, m(R ,xA) = * zz m (A) m (A) m (A) + m (A). Then for x e E, z e Z, we have <(m Z m Z )(A)x,z > = = E(1 A ) " E(1 A } = E(1 A <(B z'" B z )x,Z >) Tne Same computation gives <(m S t m S t m St + m St )(A)x,z > = O E ( 1 A <(A R (BÂ°))x,z >). Now, since p[B , ,] = B , ,, etc., we hav? zz ' s t s t o P[B Z' " B Z ] = B z' " B z' and p[A R (B )] = A R (B K In faCt (we z z ' z z ' give the proof for the first; the second is the same), for A e F, x e E, z e Z, we have Â«B z , B z )x,z >1 A = l A 1 A e l"(P) since each term is. Also, p(<(B . B )x,z >1Â„) = p(1 z z A z A 1 A ) . p( lA ) PÂ«B z x,z >1 A ) = <3 z .x,z >1 p(A) z VVVA) = <( V V^V 1 p(A) Â» S Â° p[ V " B z ] " V and the same for A D (B ). Then, by Proposition 1.5.6, both H , ZZ i Â° Â° I . Â° . B , B I and A D (B ) I. are Pmeasurable. Also, B . B I z z R zz , " z z' I B Z 'I + B  < V , + V < 2V .; hence b . B j is Pintegrable; . Â° similarly, a r (B )  < ^^.\ hence a r (B ) \ is Pintegrable as zz' '" zz' well. Also, by properties of liftings, <(B , B )x,z> and z z <(A R (B ))x,z Q > are measurable for x e E, z e Z (see property 2 77 ' U
PAGE 139
136 after Defn. 1.5.5). By the "converse" of the generalized RadonNikodym Theorem (Theorem 1.5.9), there exist measures m D : F * L(E ' Z ' ) and m R : F + L (E,Z') (the measures have the same values as the function B since Z' is a dual (cf. part 3(a) of the statement of this theorem) with finite variation m I and m I 1 D 1 ' R 1 such that: i) = E(1 A <(B . B z )x,z Q >) for A e F, x e E, z Q e Z (by taking f = x1 ft in 1.5.9), and = E(1 A <(A R (B ))x,z Q >) likewise. Also, zz ' ii) m D (A) E(1 A BÂ°. BÂ°), and ni R (A) E(1JA R (BÂ°)) zz' for A e F (we take ip = 1 in 1.5.9). From (i) we have = E(1.<(B , B )x,zÂ„>) = u U A z z z z <(m m )(A)x,z > from earlier. Likewise, = u R ,. S 't ' S 't St ' St . , . , <{ m m m + m )(A)x,z >. Both these hold for all A, x, z Q so we have m D (A)x = (m Z m Z )(A)x, and m R (A)x = m 3 m st + m st )(A)x for all A, x; hence m n = m Z m Z (m s and m m K s't' s't St' St . . . , m m + m (and in particular m , m have values in L(E,F)), By (ii), we have m Z m Z (A) = m_(A) = E(1,BÂ°. BÂ°). i S 't ' s't and similarly m m st st + m (A) E(1 A (B )) for A e F. On the other hand, we have l(m Z m Z )(A) = m(D ,xA) I ^ 1 zz " m(D zz ,xA); hence m Z m Z (A) < m(D ,xA) = m Z (A) m Z (A) = E(1 V ) E(1 V ) = E(1 (V ,V )); hence E(ljBÂ°, BÂ°l) < AZ AZ AZZ A'z Z "
PAGE 140
137 E(1.(V ,V )) for all A e F, so 3 , B I < V , V Pa.s. for a z z " z z z z each z < z'. The same computations for the rectangle yield A D (B )J < A d (V) Pa.s. If we take z,z' with rational t\ , n , ZZ ZZ o coordinates, we can find a common negligible set and modify B on it to get the inequalities everywhere for all z,z' rational. Next, lee z be fixed. We show that for any sequence r + z, r n n rational, the sequence (B (w)) is Cauchy for all w. In fact, the n sequence (V (w)) is Cauchy for all w since V is right continuous, n i.e., for any e>0, there exists n such that n,m > n implies w w K V^ (w) V (w) < e. Then, for n,m > n , we have b (w) B (w)j n m n m I v Â„ ( w ) " v r ( w ) < e Â« Thus, for any w, the sequence (B (w)) is n m r n Cauchy in L(E.Z') complete, so lim B (w) exists. n n Now, let (r ),(s ) be two sequences of rationals decreasing to n n Â° z. We can construct a sequence (v ) decreasing to z, containing subsequences of both (r ) and (s ). Then lim B (w) exists; moreover, n n v J J since lim B^ (w) and lim B (w) exist, and subsequences of both are n ' n n n contained in (3 (w)), all three limits are equal. In particular, lim B (w) = lim B (w) , so we get the same limit for any sequence of n n n n p rationals decreasing to z. Then, for every w c Q, z e R , B (w) = lim B (w) exists. The stochastic function B thus z r r + z r rational defined is right continuous.
PAGE 141
138 In fact, let e>0: there exists a neighborhood to the right of z so that if r is rational and lies inside the neighborhood, then B (w) B (w)  < . For any z* in this neighborhood, there exists a similar neighborhood for it, and the intersection of these has nonempty interior. Let r be in their intersection, r rational. We have B (w) B ,(w)  = B (w) B (w) + B (w) B ,(w) < B z (w) B r (w)J + B r (w) B ,(w) <  +  = e. Thus 3 is right continuous. Some more properties of B are the following: a) For wz, r>w. Then B 3 I = 1 q r IB B +3 B +B B I < V V. Letting q+z, we get q z z w w r 1 q r =>mÂ» b o B 3 + B B  < V V by definition of B and right continuity of V. Letting r+w likewise, we get 3 B I < V V . We ' z w 1 z w similarly have A_ (B) I ^ A n (V). zz zz o 2 8) For each z, B = B a.s.: In fact, for z e R , r rational, z z ' +' ' r>z we have JB B I S V V a.s. (in fact there is a common r z r z negligible set outside of which this holds for all r>z) letting r+z, we have b B I = liralB B I < lim(V V ) = a.s., i.e., 1 z z 1 ,'r z 1 r z r+z r+z B = B Pa.s. z z Next, we show that B has raw integrable variation B. Since B = B Pa.s., B , B = B , B a.s., and AÂ„ (3) = z z z z z z R , zz A (B ) a.s.; hence p[B , B ] B* B , and p [a (B)]
PAGE 142
139 = A R (B) (property (5) following Definition 1.5.5), so by 1.5.7, zz ' B Z ' " B  and a (B)[ are measurable; hence the finite sums we z z ' use to compute Var^^^B^ Â§ Q) ) , Var [0>t] (B (0 ^ # } ) , and Var r,A m / vnt( b ) are also measurable. L (0,0) , (s,t) J Moreover, since B is right continuous, we can compute the variation using partitions consisting of rational points: the first two terms from the onedimensional result, the third by Proposition 2.2.5. Each of these limits can then be taken along a sequence, so Var [o.s] (B (.,o) ) ' ^[o.ti^o.o*' Var [(o,o),(s,t)] (B) area11 measurable; hence Ib^ B ((J>0)  Var [0>g] (B (# >Q) ) Var [0,t] B (0,.) + Var [(0,0),(s,t)] (B) iS measu rable, i.e., B is a raw process, evidently increasing. We show now that B has integrable variation. We shall show, in fact, that b Â£ V. We proceed with each term of B separately: 1) Since Ko,o)l K v (o,o) we have l B (o,o)l * v (o,o) a s 2) Let o: = s. < s < ... < s = s be a partition of [0,s], 1 m m1 m1 We have I B. Â„. B, ,.h I (V, , V, J 10 (S i*1' 0) (3 i' 0) 10 ( V 1 ' 0) (s i' 0) (V ( S1 ,0) " V (0,0) ) + (V (s ? ,0) " V (s 1 ,0) ) + "Â• + (V (s,0) " V (s V i Â£ i mi = v ^o m " V /a n\Â» p_a 'SNow, let (a ) Â„ be a sequence of partiIs, 0) (0,0) n neN tions of [0,s] such that Var rn ,(B, _,) = lim Var r Â„ ,(B, Â„.;o ). LO.sJ (',0) [0,s] (',0) n Each term is dominated by V^^ V (Q>0) a.s., and y { ^ Q) V (0>0) does not depend on n, so taking limits we have Var r ^ ,(B, .) Â£ LP, s J (',0) \s,0) " V (0,0) a S '
PAGE 143
140 3) The same argument gives Var f (B, .) < V V 4) Let oxx be a grill on [ (0,0) , (s,t) ] : we have I a (B)J Rcoxt R^ A R (V) = \s,t) V (0,t) V (s,0) +V (0,0)To see this last equality, consider for each w the measure m , V(w) 2 on B(R + ) associated with V(w). Consider the halfopen rectangles R associated with the closed rectangles R of oxt. Then r ^ V v) . ., ( , ]( ui) Vw) Â«(o.o).(..t)]) v (5>t) v (0>t) ~ V (s 0) + V (0 0)" This holds a,s * Taking supremum along a sequence of grills as before, we have Var r .Â„ Â„. . N ,(B) < V, V [(0,0) ,(s,t)] v (s,t) (0,t) " V ( Si o) + V (0,0) 3,S " Addin 8 U P (DWi we obtain bL l B (0,0)l + Var [0,s] (B (.,0) ) + Var [0,t] (B (0,.) ) + Var [(0,0),(s,t)] (B) " V (0,0) + (V (s,0) " Vo)' + (V (0,t) " V (0,0) ) + (V (s,t)" V (s,0) " V (0,t) + V (0,0) ) = V (s,t) a ' S Â™ US ' f0r (s '^ e R + 2 ' ! 3 l (s ,t) V (s,t) a ' s Since I 8 ' is ^creasing, and B (n>n) < V ( ^ n) a.s., b is finite outside an evanescent set*; hence b is right continuous outside an evanescent set. Then, since both b and V are right continuous, b S V outside an evanescent set. In particular, l B L V oo' 30 B has integrable variation, and this completes the proof of (i). More precisely, for example is a negligible set N such that n w t N => 3 (w) < V. (w); hence b < V, n '(n,n) (n,n) ' ' (n,n) outside N n . Then N Q = {J N is negligible and B < Â• outside this set. n
PAGE 144
mi Now for (ii). For x e E, z n e Z we ha ve = z z a.s., so by (2') is integrable; moreover, is right z u z continuous since B is, and has integrable variation (we showed in the proof of 14.1.3 that  (w)  < b (w) fx  z  ) . Also, for each 2 x e R , is Fmeasurable since B is; hence is a + z u z z realvalued raw process with integrable variation, which is (ii). We now turn to assertions (1)(4). Proof of (1 ) : For M = [0,z]xA, A e F, we have, for x,z : z Â° = = E(1. (by (2')) = E(1.<3 x,z >) u uazu AzO E(1 A / 2 1 co, 2 ] d< V'V ) = e). Then l^w^.Vl = H + E(JV ; hence m(M) < E(Jl dB ). On the other hand, from B < V we have E ( / 1 M d  B  w D < E(/l dV ) = m(M); hence m(M) = E(/l M d3 w ) for M = [0,z]xA. By additivity, then, as usual, we have m(M) = E ( /l M d  B  w ) = jj . (M) for M = RxA, A t F, R a 2 rectangle of the semiring generating B(R ) (there are four kinds; cf. Theorems 3.2.2 and 4.2.1). As m, y, , are both aadditive, and sets of the form M = RxA form a semiring generating M, we have H = u 3i Now, let X be Evalued, measurable. If X e L_(m) , then E X e L (m) = L (vi B i) => X e L E ^ B ^ Conversely, if X E L E y (B)* then IM E l1( ^ j b ) => H X I E L^hl): hence X e LÂ„(m) since X is measurable. E
PAGE 145
142 We already showed that m = p.,, which establishes the second equality in (1). As for the first, we note that by Theorem 4.2.3, E(< J 2 X w dB w' Z > ' ) is defined for z E z To prove the equality, we R + note first, that by Theorem 4.2.2 there exists a stochastic measure m: M Â•* L(E,Z') corresponding to B satisfying = E(< J 2 X w dB w' Z > ' ! * ^ Note that since Z' is the dual of a Banach space, R + m has values in the same space of operators as B; cf. statement (3a) of this theorem.) We shall show that m = m; as both are oadditive it will suffice, as above, to prove for sets of the form M = [0,z]xA, A e F. Let M = [0,z]xA, A e F; let x e E, z e Z (which is a norming subspace of (Z')'). We have = = MO E() = E(/l M d) (by 4.1.3) = E(1 A ) E(1 A ) = . Then, for x e E, z e Z, = ; hence m(M) = m(M) for M = [0,z]xA. As p before, we conclude that m = m on all of B(R )xF. Then for X e L (m) we have = = E(), which t J w w completes the proof of (1). Proof of (2) : If Bx is separably valued for x e E, then Bx is measurable for x e E since it is weakly measurable by (ii). If B is separably valued, then B is measurable since is measurable for all x e E, z Q e Z, by (ii) [6, Proposition 24, p. 106]. Proof of (4) : If p is a lifting of P, we can choose B z uniquely a.s. for all z; in particular, outside an evanescent set for
PAGE 146
1^3 all z rational. Then B is determined uniquely outside this evanescent set; we already showed that p[3 ] = B for everv v e R 2 v v ' J ' + Proof of (3) : a ) . b) If one of these is satisfied, then B takes values in L(E,F); hence B takes values in L(E,F) since L(E,F) is closed in L(E,Z') for the metric topology (recall that B = lim B ). z r r + z r rational c): Assume E is separable, F e RNP. Let x e E, z e R 2 . The + measure m:F + F defined by u(A) = m Z (A)x for A e F is oadditive (since m Z (A)x < m Z (A)x so A +0 => m Z (A )  + => u(A n ) + 0; u is evidently additive) and has finite variation u S m  x  Â£ m Z flx. Then u << P since m Z << P, as we showed already. Since F e RNP, there is a Bochnerintegrable function B^ e L p such that u(A) = E(1 B^J ) for A e F. We can choose B' separably valued; therefore, we can consider F separable. More precisely, let S be a countable dense set in E. For x e S, we have lim B J ., = B v 7 a ' 5 ' In fact Â» for A &> E ( 1 Â« B ' ) = u(A) = u+z x,u x,z A x,u u rational m (A)x = m([0,u]xA)x * m([0,z]xA)x = m Z (A)x = E(1,B' ); hence A x,z B x,u "* B x,z a,S * Then ^ B x z (w): x e s Â» z national} is separable; hence {b^ z (w): x e E, z rational} is separable. For any z, B J _ = lim B' a. s., and we modify B' on the exceptional set; x,z ,j.^ x Â» u x,z u + z ' u rational we can do this and still get a RN derivative of y. Hence we can take F separable. We can then choose Z separable in F', norming for F.
PAGE 147
1411 2 Let B: R + xft * L(E,Z') be the stochastic function associated with m for this choice of Z. We have, then, for z in a countable dense subset Z C Z, and for A elf E(1, = E(<1 B' ,z >) u A x,z A x,z = ^Vx.zW ' = E(1 A ); hence = a,S ' for each z e Z ' Now, since Z Q is countable, there is a common negligible set; since Z Q is dense in Z we have B' = B x outside this negligible set. There is then a common negligible set such that 3' = B x x,z z for all z rational, x in a countable dense set of E. Then, by right continuity of B and closure of L(E,F) in L(E.Z'), we have B = B' e L(E,F) outside this negligible set, i.e., up to evanescence. By modifying B on this evanescent set, we obtain B with values in L(E,F). Moreover, since B x = B' , B x is intestable (in z x, z z particular Bx is measurable and separably valued by right continuity) and so, by 4.2.3(2), for X e L^m) , we have m(X) = E( f X dB ) t ' ; 2 v v K + (m is the measure associated with B via 4.2.3), and in particular, m(M)x = m(y) = E(/l M xdB ), which completes (c). d) Assume the range of m is contained in GCL(E.F) with G e RNP. We write G = L(R,G) and apply (c): R is separable, G e RNP. Then B has values in L(R,G) = GCL(E.F), and 8a is measurable for a e R; hence B is measurable. Also, for e L 1 (m) we have by (c) m(<{>) = Etj^dB ), which is (d), and completes the proof of this theorem. m Remark . In the last part, once we have B measurable, we can also get the equality by applying Theorem 4.2.1.
PAGE 148
145 4.4 On the Equality J m j = y. , In Theorem 4.2.3, we began with a stochastic function B with integrable variation, and associated a measure m with finite variation, and we proved that m < y. .. We now consider some cases where this is in fact an equality. Theorem 4.4.1 Let E,F be two Banach spaces and Z C F' a subspace norming for F. Let B: R^xP. > L(E,F) be a right continuous stochastic function satisfying conditions (i) and (ii) of Theorem 4.2.3, and let ra: M Â» L(E,Z') be the corresponding measure with finite variation m satisfying = E() V V V for any Evalued measurable process X e L^y, ,). We have the E B equality ]m = y, . , i.e., IH( X I> = E(J 2 lX v d3J v ) for X e Lg(a) R + in each of the following cases: 1) There is a lifting p of P such that p[B ] = B for z z z c R . + 2) E is separable and there is a countable subset SCz norminj for F. 3) E is separable and B x is integrable for every x e E and Â„2 z e R . + 4) B is measurable and B is integrable for z e r 2 .
PAGE 149
146 Proof.' Ca se (4) has been dealt with in Theorem 4.2.1; we include it here for completeness. We have a measure m: M > L(E.Z') with finite variation. By Theorem 4.3.1, there exists a stochastic function B' with values also in L(E,Z') (by 3(a)) satisfying (i) and (ii) of Theorem 4.3.1, such tnat p[B'] = B' for z g R and such that for every X e L 1 (m) we have z z + jr (X),z > EÂ«/ 2 X v dB;, Z() >) for z Q e Z (not_e: here we consider Z embedded in Z" as a norming subspace in order to apply the theorem), and IÂ»I<X> = E(J X dB' ). R Now, for X = i M x with M e M, x e E, and for z e Z , we have E(/l M d) = E(/l M d). In fact, E( Jl M d ) = E ^) (by 11J ' 3) = E(< / X v dB vÂ» Z >) = z > = E() = E() = E(/l M d). Taking M = [0,z]xA with A eFwe obtain '^[O.zlxA^v*^ = ^V^'V 5 ' and ^Â•Ko.zJxA^V'V) = E(1 A ): hence E(1 A < B ;x,z Â» E(1 A ) for A ef so (1.1) = a.s. for x e E, z e Z. We shall prove that in cases (1)(3), B and B' are
PAGE 150
147 indistinguishable; hence B' = b up to evanescence, so IB'I = l m l= y 1) Assume p[B z ] = B^ for all z e R^. We have, from above, p[B^] = p[B z ] for z e R + as well. Then from (1.1) we conclude that B z = B z a ' 3, for each z ( P r Â°P ert y ^ following Defn. 1.5.5). Since both are right continuous, they are indistinguishable. 2) Assume E is separable, let E Q C E be a countable dense set, SCZ a countable subset norming for F. We have = z a,S * for ail x G E o' z e S " There is then a common negligible set N such that the equality is valid for all z e R 2 + rational, x e E , z Q z S. By right continuity, then, this holds 2 outside N for all z e R . Since S is norming, we have B'x = B x + z z 2 outside N for all z e R + , x e E Q . Since E Q is dense in E, we have , 2 B x = B x for all z e R x t E (still outside N 1 ) ; hence 3' = B c z + z z outside the evanescent set R + xN, i.e., B and B' are indistinguishable. 3) Assume now that E is separable and 3 x is integrable for z 2 every x c E, z e R + . Then B z x is almost separably valued; by right continuity Bx is separably valued outside an evanescent set for x z E. Since E is separable there is (as before) a common evanescent set A outside of which Bx is separably valued for all x e E. We modify B on A by setting it equal to zero on A and get a process B" indistinguishable from B, with B" taking values in a separable space F Q C F for all x e E. men E(/l M d) = E(/l M d) = ;
PAGE 151
118 hence m is the measure associated with the stochastic function 2 B": R + xn * L(E,F ). Since F is separable, there is a countable subset SCZ norming for F . By (2), we have m(M) = E( fl dB" ). U ' ' J M ' ' v Since B = B" outside an evanescent set, b = B" outside an evanescent set, so J 1 d  B I = / _1 u dlB" a.s. for any M e M; ; 2 M ' ' v ; D 2 M ' ' v R + R + hence m(M) = E(fl dB" ) = E ( f 1 d I B I ) for M e M, i.e., ' M ' ' v Â•'M' l v  m  = ui R , and this completes the proof. I Remark . If we start with a stochastic measure and associate a function, we always have m = u,_i, but if we start with a stochastic l B l function, we do not get equality Â— not even if the measure has values in L(E,F). Equality (1.1) seems to be as close as we can come in general; in order to get everywhere from there, it seems we need for E and Z not to be "too large."
PAGE 152
CHAPTER V CONCLUSION We have seen that the usual definition of the variation on a rectangle of a function of two variables is insufficient to yield all the properties necessary to extend the theory of Stieltjes measures to functions of finite variation on the plane. We have given some additional conditions sufficient to establish a proper definition of the variation of a function, and although these were not shown to be minimal, it would seem to be difficult to weaken them further. We have shown that, starting with a twoparameter stochastic function X with values in L(E,F), we can associate a measure u with x values in L(E,Z') and that under certain conditions u has values in x L(E,F) as well. We have also established a similar correspondence, starting with a measure and obtaining a stochastic function. We have also shown that, if the spaces E and F are not "too large," we have the equality Ki = M xWe hope that this lays the groundwork for exploring the question of existence of optional and predictable projections of vectorvalued multiparameter processes. 1 49
PAGE 153
BIBLIOGRAPHY 1) Bakry, D., Limites "Quadrantales" des Martingales , in Processus Ale"atoires a Deux Indices , Colloque ENSTCNET, Lecture Notes in Mathematics no. 863, SpringerVerlag, New York, 1980, pp. 4049. 2) Cairoli, R., and Walsh, J.B., Stochastic Integrals in the Plane , Acta Math., 134(1975), pp. 111Â—1 83~~ 3) Chevalier, L., Martingales Continue a Deux Parametres , Bull. Sc. Math., 106(1982) , pp. 1962. 4) Dellacherie, C, and Meyer, P. A., Probabilities and Potential B , chaps. IIV, NorthHolland, New York, 1978. 5) Dellacherie, C, and Meyer, P. A., Probabilities and Potential , chaps. VVIII, NorthHolland, New York, 1982. 6) Dinculeanu, N., Vector Measures , Pergamon Press, New York, 1967. 7) Dinculeanu, N., VectorValued Stochastic P r ocesses I: Vector Measures and VectorValued Stochastic Processes with Finite Variation , Journal of Theoretical Probability (to appearl. 8) Dunford, N., and Schwartz, J.T., Linear Operators, Part I: General Theory , Pure and Applied Math., no. 7, WileyInterscience, New York, 1958. 9) Fouque, J. P., The Past of a Stopping Point and Stopping for TwoParameter Processes , Journal of Multivariate Analysis. 13(1983), pp. 561577. 10) Kussmaul, A.V., Stochastic Integration and Generalized Martingales , Pitman, London, 1977. 11) Me'tivier, M. , Semimartingales , Walter de Gruyter, Berlin, 1982. 12) Meyer, P. A., The'orie Ele'mentaire de Processus a Deux Indices in Processus Altjatoires a Deux Indices , Colloque ENSTCNET. Lecture Notes in Mathematics no. 863, SpringerVerlag, New York, 1980, pp. 139. 13) Millet, A., and Sucheston, L., On Regularity of Multiparameter Amarts and Martingales , Z. Wahr . , 56(1981), pp. 2145. 1 50
PAGE 154
151 11) Neveu, J., Discrete Parameter Martingales , NorthHolland, New York, 1975. 15) Nualart, D., and Sanz, M., The Conditional Independence Property in nitrations Associated to Stopping Lines in Processus Aldatoires a Deux Indices , Colloque ENSTCNET, Lecture Notes in Mathematics no. 863, SpringerVerlag, New York, 1980, pp. 202210. 16) Radu, E., Mesures Stieltjes Vectorielles sur R n , Bull. Math, de la Soc. Sci. Math, de la R.S. de Roumanie, 9(1965), pp. 1 291 36. 17) Rao, K.M., On Decomposition Theorems of Meyer , Math. Scand., 21(1969), pp. 6678. 18) Walsh, J.B., Optional Increasing Paths in Processus Ale~atoires a Deux Indices , Colloque ENSTCNET, Lecture Notes in Mathematics no. 863, SpringerVerlag, New York, 1980, pp. 172201.
PAGE 155
BIOGRAPHICAL SKETCH Charles Lindsey was born in Lexington, Kentucky, on April 7, 1962, and lived there until 1969 when his family moved to Merritt Island, Florida (where his parents still live). He graduated from Merritt Island High School in June, 1979. He began his undergraduate career in fall 1979 at the California Institute of Technology and stayed there until December, 1980, when he transferred to Auburn University. He attended Auburn the first six months of 1981, then transferred to the University of Florida and has been there until the present. He received his B.S. degree from the University of Florida in April, 1983; his M.S. in December, 1984; and expects to receive his Ph.D. in April, 1988. 152
PAGE 156
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ( / / .' U i L Nicolae Dinculeanu, Chair Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. James Brooks Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. wi SCUlJ, cuccr\/^ Louis 31ock Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Joseph Glover Professor of Mathematics
PAGE 157
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. James Keesling Prjofessor of Mathematics 1 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ftjdL^ Â£&iÂ£AJ> William Dolbier Professor of Chemistry This dissertation was submitted to the Graduate Faculty of the Department of Mathematics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. Dean, Graduate April, 1988 M~*s**>UÂ£*r^*^ > <^J^^U^l~*^C~ School
PAGE 158
iimiuPBSITY OF FLORIDA liliniuif 3 1262 08553 2017
xml version 1.0 encoding UTF8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EZJ2KM4BH_9CV8KL INGEST_TIME 20170720T21:07:04Z PACKAGE UF00099467_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
TWOPARAMETER STOCHASTIC PROCESSES
WITH FINITE VARIATION
BY
CHARLES LINDSEY
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
TABLE OF CONTENTS
Page
ABSTRACT iii
CHAPTERS
I INTRODUCTION 1
1 .1 Notation 3
1.2Filtrat ions 9
1 .3 Stochastic Processes 8
l.ii Stopping Points and Lines 8
1.5 Some Measure Theory 12
II VECTORVALUED FUNCTIONS WITH FINITE VARIATION 21
2.1 Basic Definitions and Some Examples 21
2.2 The Variation of a Function of Two Variables 26
2.3 Functions of Two Variables With Finite Variation *43
III STIELTJES MEASURES ON THE PLANE 69
3.1 Measures Associated With Functions 69
3.2 Functions Associated With Measures 85
IV VECTORVALUED PROCESSES WITH FINITE VARIATION 95
H. 1 Definitions and Preliminaries 96
ij.2 Measures Associated With VectorValued
Stochastic Functions 112
i).3 Vectoiâ€”Valued Stochastic Functions Associated
With Measures 131
i).i) On the Equality m  UB 199
V CONCLUSION 199
BIBLIOGRAPHY 150
BIOGRAPHICAL SKETCH ....159
ii
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
TWOPARAMETER STOCHASTIC PROCESSES
WITH FINITE VARIATION
BY
CHARLES LINDSEY
April, 1988
Chair: Nicolae Dinculeanu
Major Department: Mathematics
2
Let E be a Banach space with norm  â– , and f: R+ â– + E a function
with finite variation. Properties of the variation are studied, and
an associated increasing realvalued function f is defined.
Sufficient conditions are given for f to have properties analogous to
those of functions of one variable. A correspondence f p between
2
such functions and Evalued Borel measures on R is established, and
+â–
the equality p^, = Pf is proved. Correspondences between Evalued
twoparameter processes X with finite variation ]Xj and Evalued
stochastic measures with finite variation are established. The case
where X takes values in L(E,F) (F a Banach space) is studied, and it
is shown that the associated measure p^ takes values in L(E,F"); some
sufficient conditions for p^ to be L(E,F)valued are given. Similar
results for the converse problem are established, and some conditions
sufficient for the equality p^ = p
are given.
CHAPTER 1
INTRODUCTION
Families of random variables indexed by directed sets have been
objects of study for some time. The most common by far, though, has
2 2
been R , and especially r the first quadrant of the plane, as this
is considered the most "natural" extension of the usual indexing set R
or R . One of the principal objects of study relating to such
processes has been the stochastic integral of processes indexed by the
plane.
Stochastic integration with respect to twoparameter Brownian
motion has been studied extensively; the focus of more recent study
has been the more general problem of extending the oneparameter
theory of stochastic integration with respect to a semina"tingale. In
one parameter, this is done by writing a semimartingale X as
X = M + A
with M a locally square integrable local martingale and A a process of
finite variation (see, for example, Dellacherie and Meyer [5]). The
major problem in two parameters is with the notion of "local martingale"
the theory of stopping is not sufficiently welldeveloped to permit a
definition with all the necessary properties. Some preliminary work
has been done, however, using the notion of an increasing path (see,
for example, Fouque [9] and Walsh [18]). A definition of a stopped
1
2
process has also been given for square Integrable martingales in Meyer
[12], but this too is rather limited.
Another area that has seen renewed interest is the study of
processes with values in a vector space, especially in a Banach space
(see, for example, Neveu [1A]). In Dinculeanu [7] the correspondence
given in Dellacherie and Meyer [5] between processes with finite
variation and stochastic measures Is extended to the case where the
processes have values in a Banach space, and in Meyer [12] the
correspondence is stated for twoparameter realvalued processes. In
the oneparameter theory, this result finds its use in applications to
projections, what in turn (at least in the scalar case) Is relevant to
the decomposition of supermartingales and to semimartingales. This
correspondence X <* is given by
(1.1.1)
for 4> scalarvalued, bounded process. We shall extend this
correspondence to the case where X has integrable variation, with
values in a Banach space E.
Since the inner integral on the right side of (1.1.1) is computed
pathwise, we begin by studying the properties of Banachvalued
2
functions f: R+ â– * E with bounded variation. In Chapter II, we
develop the relevant properties of such functions, and in Chapter III
2
we show that to each such f we can associate a measure u : B(R+) â– * E
with finite variation, and we prove the equality
(1.1.2)
3
(i.e., the variation of the measure associated with f is equal to the
measure associated with the variation of f). In Chapter IV we prove
that a stochastic measure can be associated with a process of
integrable variation, in the same manner as in Dinculeanu [7], and we
consider the converse problem: that of associating a function with a
given stochastic measure. Unfortunately, the equality (1.1.2) is not
preserved in general for processes and stochastic measures, so we end
up by establishing some sufficient conditions for the equality to hold
1.1 Notation
2
The index set is R+; we shall sometimes consider functions and
2
processes extended by zero to all of R . In the rare cases where we
look at points outside the first quadrant, we shall say so
2
explicitly. We denote points in R+ by z, u, w, v and their
coordinates by the letters r, s, t, p. For example, then, we write
z  (s,t) , u  (p,r), etc.
There are two notations commonly used in the literature for the
2
order relation in R+: we adopt here the notation of Meyer [12], For
two points z = (s,t), z'  (s',t'), we have zÂ£z' iff sSs', tSt';
z
z*z'.)
We denote by (z,z'] the set of all u such that z
define analogously the rectangles [z,z']( [z,z'), and (z,z'). We
* In their pioneering paper, Cairoli and Walsh [2] use "<" in place
of <, and "<<" for the strict inequality. This is used some in
later places (e.g., in Walsh [18]), but the notation we adopt seems
to be more common now.
i)
denote [z,*) â– = {u: zSuj, and by FÂ¡z the rectangle [0,z] (or (Â°Â°,z] lf
2
we are discussing all of R ; these cases will be stated explicitly
when they occur). We shall also have need to state the coordinates of
rectangles explicitly: we write, for example, (z,z'] * ((s ,t), (s',t') ]
(  (p,r): s
2 2
For a function f defined on R+ (or R ), and a rectangle
R = (z,z'] = ((s ,t) , (s',t') ], we define the (rectangular) increment of
f on R, denoted A , (f) by
zz
A ,(f) = f (s ' ,t')  f (s ', t)  f(s.t') + f(s,t).
zz
The notation A ,(f) is used for this sum regardless of whether the
zz
rectangle is open or closed.
1.2 Flltrations
Let (n, F,P) be a complete probability space. There are two
different methods in the literature for constructing flltrations
2
on R+; we shall give both.
The first method is the one adopted by Meyer [12], We begin with
two flltrations satisfying the usual conditions:
(F1) _ , also denoted (F ), and
s stR+ sÂ°Â°
(F2) _ , also denoted (F ).
t teR+ *t
2 1 2
When we wish to discuss all of R , we extend these by taking F , Ft
to be the degenerate afield for s<0 or t<0. We also make the
5
12 11
convention F = F = F (which may or may not equal F = VF or
oo co w S
2 2 S
FÂ¿ = VFP.
"" t 1
^ 1 P
We then set F = F Pi F , and verify that this family satisfies
st s t
the usual conditions.
1 2
1)Each F contains the Pnegligible sets, since F , F contain
st st
them for all s,t.
2) If (s,t) S (s',t') , then F C F , ,.
st s t
11 2 2
In fact, F C F , since sis', F. C F , since tit , hence
s s t t
Fst = F!nFtCFl'nFt' = Fs't"
3) We have F O F , (Note: In two parameters,
3t f Â» , / * * \ st
(s,t)<(s ,t )
our usual definition of right limit is for (s,t) Â£ (s',t'),
(s,t) * (s',t')> as we shall see later. The statement we
shall prove is somewhat stronger.)
Proof. One containment is evident: since F C F , , for
st s t
each (s',t') > (s,t) by (2) , F C C\
(s,t)<(s\t')
For the other containment, let
A e n F = n (Fl,n F* ).
(s,t)<(s',t') s'>s
t '>t
Then A e F1 , for all s'>s, hence A e F^, = f\
s '>s
Similarly, A e F^, for all t'>t, hence A a F^, = F^
t
Then A e (F1 Pi F^) = F , hence F D ( O F , ,).
s t st st s t
s '>s
t '>t
Putting the two containments together gives the equality.
6
We say that the condition of commutation is satisfied if the
1 2
conditional expectation operators E( â€¢ I F ) and E(â€¢IF ) commute,
1 s t
12 2 1
i.e., if E( â€¢ I F F )  E( â€¢ I F F ). The product is then equal to
1 s' t ' t1 s
E( â€¢ I F1 = E( 1 F . ). (In fact, denote, for f e l2(P), E a Banach
1 3 t 1 St E
12 2 1
space with the RNP, g = E(f I F F.) = E(fF If ). Then g is measurable
â€™ s1 t â€™ t s
2 i
with respect to F and F , hence g is F measurable. Also, for
t s st
A e Fst> we have JAgdP = /A E (r  F^  F^)d P = /flE(fF^dP = JflfdP
since A e F = F1 nF2. Thus, g = E( f  F ).) All of the main
st s t st
results of the theory of twoparameter processes require the condition
of commutation.
The second (and more frequently used) way of describing
p
nitrations on R+ is due to Cairoll and Walsh [2]. There, we begin
with a family (f , z e R^l of subofields of F satisfying the
following axioms:
(F1) If zÂ£z', then F CT F' (this is (2) of Meyer),
z z
(F2) F contains all the negligible sets of F. (Note: This,
o
along with (Fl), implies condition (1) of the Meyer
construction .)
(F3) For each z, F  F (This was condition (3) of
z z
z >z
Meyer.)
1 2
We now define F = F =VF.,Ft=F.=VF . In place of
s sÂ® t st t Â»t s st
the condition of commutation, we impose the axiom
1 2
(F*0 For each z, F and F are conditionally independent given
z z
1 2
F , i.e., for Yâ€ž F measurable, integrable, F 
z 1 z 2 z
measurable, integrable, we have
7
(1.2.1) E( Y Y IF ) = E( Y  F ) E( Y  F ).
This condition is equivalent [4, 11.115] to
(1.2.2) e(Ã If2) = e(y If )
1 1 z 1 1 z
for every F1measurable, integrable r.v. Yâ€ž. The condition (Ft) can
z â€” 1
be seen to be equivalent to the condition of commutation as follows:
(Ft) => commutation: Let X be Fmeasurable, integrable. Then
E(XF^) is F^meas unable, integrable. By (Ft), E( E(X  fâ€™ )  F2) =
E( E( X I F1) I F ) = E(XF ). Similarly, E(xF2) is F2measurable,
1 z 1 z 1 z 1 z z
integrable; hence by (Ft) E( E (X  F2) ] F1) = E(E(XF2)F ) = E(X  F ).
z z z z z
Thus E( E(X [ F1 )  F2) = E(X  F ) = E ( E (X  F2)  F1 ) , i.e., the operatorâ€™s
z z z z z
E(â€¢IF2) and E( IF1) commute.
1 z 1 z
Commutation => (Ft): He shall assume commutation and prove that
(1.2.2) holds.
Let Y be F^measurable, integrable. We have, from the
commutation condition,
E(Y1Fz) = EiYjF^F2)  EiYjF2),
which is just (1.2.2).
Although the difference between the two constructions is slight
(the main difference being on the "border at infinity," where we do
not necessarily have right continuity of the filtrations (F^) and
p
(F~) in the CairoliWalsh model), the condition most often imposed on
the filtration in the literature is stated as the (F4) condition,
although the notation used is more often that of Meyer.
1 .3 Stochastic Processes
Throughout this section, (n,F,P) is a complete probability space,
(F ) a filtration satisfying axioms (FI)(FA) (in particular, one
Z zeR^
constructed by the method of Meyer), and E will denote a Banach space
equipped with its Borel ofield, denoted B(E). The definitions in
this section are taken from [12], [7].
Definition 1.3.1. A stochastic process is a measurable (i.e., a
(B(R^)xF, 8(E))measurable) function X: R^xR + E. We usually denote
X(z,w) by X^iw), and the mappings w *â– Xz(w) by Xz .
Remark. It will sometimes be convenient to extend the index set to
2
all of R by defining Xz ' 0 for z outside the first quadrant, and
Fz  the degenerate ofield for those z. When we wish to consider a
process in this light, we shall say so explicitly.
A brut or raw process is a process X such that is Fmeasurable
2
for all z e R . X is called adapted if X is F measurable for all
+ z z z
02
z e R+.
2
A process X is called progressive if, for every z e R+, the map
(z,w) â– + Xz(w) from [0,z]x!2 â€¢> E is (B( [0,z])xFz> B( E) )measurable.
(Note: This definition comes from Fouque [9]; in Meyer [12]
progress!vity is defined using halfopen rectangles [0,z). The
definition we give here is the one in common use today.)
1.A Stopping Points and Lines
1.A.1 Stopping Points
The notion of stopping plays a fundamental role in the theory of
oneparameter processes. The obvious extension of the definition of
9
stopping time to twoparameter processes yields what is known as a
stopping point.
2
Def ini ti on 1.i. 1 . A stopping point is a mapping Z: Q * R + such that
o
for all v e R^, the set (z Â£ v) belongs to F .
We often denote Z = (S,T). The components S,T are, it turns out,
1 2
stopping times with respect to (Fg), (Ft), respectively, but this by
itself is insufficient to guarantee that (S,T) is a stopping point.
We do, however, have the following oharacterization [12]:
1 2
Theorem I.H.2. Let S be an (Fg)stopping time, T an (Ft)stopping
2
time. Then Z = (S,T) is a stopping point if and only if S is F^,
measurable, T F^me asura ble.
Although stopping points have found some application recently
(see, for example Walsh [18] and Fouque [9]), they are rather
inadequate for the purpose of developing a theory of localization for
twoparameter processes. To begin with, due to the partial order
in R , we are not even assured that z>v( c F ! Also, if U and V are
stopping points, UV may not be. Moreover, in one dimension, the graph
of a stopping time T divides R+xil into two components, namely the
stochastic intervals [[0,T)) and [[T,Â«0), whereas the graph of a
stopping point Z does no such thing. Furthermore, an important
realization of a stopping time is as the debut of a progressive set,
and we have no analogous result in the plane.
1.^.2 Stopping Lines
1 0
O
Let ACR xi! be a random set (the usual definition: 1 , is a
measurable process). The open envelope of A, denoted (A,Â®), is the
random set whose section for each w e fi is given by
(A,Â®)(w)  (z,Â®).
zeA(w)
Some properties of (A,Â») (proofs can be found in Meyer [12]):
1) If A is progressive, (A,Â®) is predictable.
2) The Interior of a progressive set A is progressive, from which
we obtain, by passing to complements, that the closure of A is
progressive.
Vie designate in particular by [A,Â») the closure of (A,Â®) (i.e., for
each w, we define [A,Â®)(w)  (A,Â®)(w)). [A,Â®) is progressive if A is
progressive, by the above properties; it is called the closed envelope
of A. The random set
D  [A,Â®) \ (A,Â®)
A
is called the debut of A: it is progressive if A is progressive.
Definition 1.4.3. a) A random set Z is called a stopping line if it
is the debut of a progressive set, i.e., if there is a progressive set
A such that
Z  D, = [A,Â») \ (A,Â®).
A
b) A stopping line Z is predictable if it belongs (as a set) to the
predictable ofield P.
Remarks.
1) The set A in the definition is not unique: for example, A
and (A, <Â°) always have the same debut.
2) This definition has drawbacks: for example, is not
necessarily adherent to A.
3) For an alternate way of defining stopping lines (as a map from
2
!J into a certain set of curves on r^), see Nualart and Sanz
[15].
*0 Since (A,Â«) is predictable, to say that a stopping line
Z = is predictable amounts to saying that [A,â€) is
predictable, which is an alternate way of defining predictable
stopping times in one parameter (cf. [A, IV.6931.
We introduce a partial order on the set of stopping lines by defining
H < K if (H,Â«) 3 (K,).
We also make the convention
H < K if (H, â€œ) 3 [K,â€œ) .
The set of stopping lines is then a lattice for Â£:
HK is the debut of (H,Â») (K,Â°Â°) , and
HK is the debut of (H,Â°0 O (K,â€).
We say that a stopping line H is the limit of an increasing (resp.
decreasing) sequence (H ) of stopping lines if [H,Â») Â» ,Â»)
n
(resp. if (H, Â°Â°) = l^J(H ,Â»)). In the first case, if we have
n
[H,Â°Â°)  rWH we say that the sequence (h ) announces (foretells)
n
H; If there exists a sequence (tO announcing H, we say H is
announceable (foretellable).
In the oneparameter theory, this is equivalent to being
predictable (and in fact many authors define predictability in this
fashion). However, we only have one implication for stopping lines,
namely, that every announceable stopping line is predictable. In
fact, if H is announceable, then ex. (H ) such that [H,Â»)  0(9 ,â€).
n
Each (H ,Â«) is predictable, hence the Intersection is predictable. By
remark (4), this implies that H is predictable. Unfortunately, the
other implication does not hold. For a counterexample, see Bakry [1].
Finally, we note the existence of a predictable crosssection
theorem for stopping lines. The proof Is essentially the same as for
the onedimensional case. We denote by it the projection
2
of R+xÃÂ¡ onto II.
Theorem 1, H. it (predictable crosssection theorem). Let A be a
predictable set, and let e>0. There exists a compact* predictable set
K satisfying the following:
1) K C A and P{K = 0, A * 0} < e
2) D is announceable and K C D .
K
1.5 Some Measure Theory
In this section we collect some results from measure theory which
we shall make frequent use of later.
That is, the section K(w) is compact for each w.
1 3
5.1 Monotone Class Theorems
There are several versions of monotone class theorems, both for
families of sets and for functions. The two stated here are the
variants we shall use later. The statements are taken from
Dellacherie and Meyer [Â¿4].
Theorem 1.5.1. Let C be a ring of subsets of !). Then the monotone
class M generated Dy C Is equal to the oring generated by C. If C is
an algebra, then M = o(C).
Theorem 1.5.2. Let H be a vector space of bounded realvalued
functions defined on !5, which contains the constants, is closed under
uniform convergence, and has the following property:
for every uniformly bounded increasing sequence (f ) of
positive functions from H, the function f = lim f belongs
n
to H.
Let C be a subset of H which is closed under multiplication. The
space H then contains all bounded functions measurable with respect to
the ofield o(C).
The most frequent application of this theorem comes when we wish
to prove that a certain property holds for all bounded Fmeasurable
functions; it allows us to reduce to the case where f is the indicator
of a set. We shall see numerous examples of this theorem at work.
1.5.2 Liftings
Let (T.E.p) be a measure space, pÂ£0. Material for this section
comes from Dinculeanu [6, pp. 199216]. We shall need these
properties in Chapter IV.
DefinÃ tions 1. 5. 3. * Let u be a positive measure.
A mapping p: lâ€(u) â– Â» l"(p) Is said to be a 1 Iftl.ng of L*(y)
if it satisfies the following six conditions:
1) p (r) = f pa.e.
2) f  g ua.e. implies p(f)  p(g)
3) p(of + Bg)  ap(f) + Bp(g) for a,S e R
4) f>0 implies p(f)>0
5) f(z) = a implies p(f)Cz) = a
6) p(f g)  pCf )p(g).
We say that L (p) has the lifting property if there exists a lifting p
of L (p). The following theorem affirms that, for probability
measuâ€es P in partÃculaâ€™', lâ€œ(p) always has the lifting property.
Theorem 1.5.4. If ir has the direct sum property, then LÂ°(p) has the
lifting property.
Let, now, E and F be Banach spaces, T a set, and Z a subspace
of F', norming for F, i .e ., such that
y  sup I Â«y jZ> I for every yeF.
' zeZ I!zIf'
For every function U: T > L(E,F) (continuous linear maps E * F) ,
x: T * E and z: T + Z, we denote by the map t *
and by â€™J  the map t * U(t)J.
* The definitions and theorems in Dlnculeanu [6] are given in
somewhat more generality; we restrict ourselves here to statements
involving the measures and ofields we shall be working with.
15
For two functions ,11^: T * L(E,F) we shall write U1 *
= pa.e. for every xeE, zeZ.
Let p be a lifting of lâ€(p).
Definition 1.5.5. Let l): T + L(E,F) be a function.
a) We shall write p(U) = l) if for every xeE and zeZ we have
e L (p) and
p() =
b) We shall write p[U]  U If there exists a family A of subsets
of T such that m has the direct sum property with respect to
A, such that for every Ac A, xcE, zeZ, we have
1^ c L (u) and
p(1 .) = 1 ....
A p ( A)
(Note: If p is ofinite (and in particular If p is a probability
measure), the relation in (b) holds for all A measurable.) The
functions U with p(U) = U or p[U]  U have the following properties:
1) p(U)  U implies p[U] = U.
2) If p[U] = U, then is pmeasurable for every xcE,
zeZ.
3)
If U
III
c:
ro
P(U,)
 u,
p(u2) = u2
, then
U1  V
A)
If u1
= U2.
p[u,]
â– U1
and p[U ] =
â– U2>
then U1 Â» U2 pa.e
5)
If U
â– U2
pa.e.
and
pCu,] = U1 ,
then
pCu2] = u2.
We shall also make use of the following:
16
Proposition 1.5.6. Let U: T + L(E,F). If p[U] = U, then the
function t + U(t)J Is pmeasurable. It is this proposition, along
with property (5) above, that will be used most often later.
1.5.3 RadonNlkodym Theorems
We state here some generalizations of the RadonNikodym theorem
to vectoiâ€”valued measures with finite variation. These pa'ticular
statements are taken from Dinculeanu [6, pp. 26327^]. Throughout
this section, T denotes a set, R a ring of subsets of T, E and F
Banach spaces, and Z a subspace of F', norming for F.
Theorem 1,5.7. Let m: R + L(E,F) be a measure with finite variation
p. If p has the direct sum property, then there exists a function
U : T * L(E,Z') having the following properties:
m
1) U (t) = 1 pa.e.
* m
2) For all f e L^,(m) and zeZ, is pintegrable and we have
E m
 [dp.
J J m
3) If p is a lifting of L (p), we can choose uniquely so
that p(U ) = U (cf. Definition 1.5.5).
m m
y) We can choose U (t) E L(E,F) for every t e T, in each of the
following cases:
a) F  Z'
b) There exists a family A covering T such that p has the
direct sum property with respect to A , such that for
every A e A , x e A, the convex equilibrated cover of the
17
set (JAi(jxdm: tp Rstep function, J tidu < 1} is
relatively compact in F for the topology o(F,2).
c) For every x e E, the convex equilibrated cover of the set
l/txdm: \i Rstep function, J"  ip Â¡ d y S l} is relatively
compact in F for the topology o(F,Z).
Note: If m is defined, on a aalgebra F, which will be the case in
our uses of this theorem, then the condition that m have the direct
sum property is automatically satisfied, and we can replace the family
by F in part (Ã¼b) of the statement.
These remarks also hold for the following generalization of the
RadonNi kodym Theorem.
Theorem 1.5.8 (Extended RadonNikodym Theorem). Let v be a scala
measure on R and m: R + L(E,F) a measure with finite variation p. If
v has the direct sum property and if m is absolutely continuous with
respect to v, then there exists a function V : T + L(E,Z') having the
m
following properties:
1) The function V  is locally* vintegrable and
Jidp = J"V Â»(KJ v  for t/ e L (p)
(here u denotes the variation of v).
2) For f e blip) and z e Z, is vintegrable, and we have
Cj ro
< if dm ,z> = idv(t).
1 â€˜ m
As before, if the measures are defined on an oalgebra, this can be
dropped.
3)
18
If p Is a lifting of l"(u). we can choose uniquely valmost
everywhere such that p[V ] = V (of. Definition 1.5.5). If,
m in
in addition, there exists a>0 such that y Â£ a v Â¡ , then we can
choose V uniquely such that p(V ) = V .
m m m
14) We can choose V (t) e L(E,F) for every t c T, in each of the
m
following cases:
a) F  Z'
b) There exists a family A covering T such that v has the
direct sum property with respect to A such that for every
A e A , x e E, the convex equilibrated cover of the set
ij^ijixdm: pRstep function, JA ii d v  Â£ l}
relatively compact in F for the topology a(F,Z).
b') The same statement as (b), with A such that y has the
direct sum property with respect to A and with
11 ij)dy Â£ 1 instead of J i>dv Â£ 1. In this case we may
not have p[V ] = V .
m m
c) For every x e E, the convex equilibrated cover of the set
(J^xdm: ij Rstep function, J  vp  d  v  Â£ 1 1 is relatively
compact in F for the topology o(F,Z).
c') The same condition as (c) with Jij)dy Â£ 1 instead of
/1 ipd v  Â£ 1. In this case we may not have p[v ] = Vra<
Theorem 1.5.7 gives a "weak density" of a vector measure m with
respect to its variation y, whereas Theorem 1.5.8, more generally,
gives such a density of m with respect to a scalar measure v not
obtained from m. The former is a particular case of the latter, but
19
we shall make good use of It in its own right, and so we take the time
to state it separately here.
The final result we shall need is a "converse" of these theorems.
Theorem 1.5.9. Let v be a scalar measure on R and U: T â– * L(E,F)
a function such that â€™J  is locally vintegrable and the function
is vmeasurable for every x e E and z e Z.
Then the function is vintegrable for f z Lg(U]v)
and z e Z and there exists a measure m: R â– * L(E,Z') such that
= dv for f z L^(uv) and z z Z,
and ijdu i /uipdv for ij e L1(ll[u).
The measu"e m has values in L(E,F) in each of the following cases:
a) F = Z '
b) For every x e E there exists a family A such that v has
direct sum property with respect to A , such that for every
A e A the convex equilibrated cover of the set {U(t)x; t e a}
is relatively compact in F for the topology o(F,Z).
c) For every x t E, the convex equilibrated cover of the set
{U (t) x: t e T} is relatively compact in F for the topology
o(F.Z) .
d) The function t >â– U(t)x is vmeasurable for each x e E; in
particular if F is separable.
If v has the direct sum property, we have the equality
u = U   v! , hence
20
Ji(jdij = J u   411 d  v  for ip e L1(u)
in each of the following cases:
Ð°) There exists a lifting p of L (p) such that o[U]  U.
B) E is separable and there exists a countable norming subset
S C Z.
v) E is separable and the function t * U(t)x is vmeasunable for
every x e E.
Ð±) The function U is vmea3urable (in this case we do not need
the direct sum property on v) .
CHAPTER II
VECTORVALUED FUNCTIONS WITH FINITE VARIATION
Since the stochastic integral with respect to a process of finite
(or integrable) variation reduces to taking the Stieltjes integral
2 2
pathwise, it is appropriate to study functions defined on R+ (or R )
with finite variation as a starting point. Throughout this chapter, f
2
will denote a function defined on R+ with values in a Banach space E,
unless explicitly stated otherwise. We shall write f(z) or f(s,t)
interchangeably for z = (s,t).
2.1 Basis Definitions and Some Examples
For functions of one variable, in order to associate an o
additive Stieltjes measure, we need the function to be either right or
left continuous. Here we shall use right continuity. In two
dimensions, however, there are two different notions of right
2
continuity: one for the order in R+, the other a condition merely
sufficient to ensure oadditlvity of the associated measure.
2
Definition 2.1,1. Let f: R+ Â» E be a function.
a) We say that f is right continuous (in the order sense) if, for
2
all z Â£ R+, we have
f(z)  11m f(u), or equivalently lim f(u)  f(z) = 0.
u*z u>z
uÂ£z u5z
21
22
(Note: We shall use sometimes the notation u + z for u Â» z,
u 2 z.)
b) We say that f is incrementally right continuous if, for all
2
z e R+, we have (denoting z'  (s',t'))
lim  A ,(f) = lira f(s',t')f(s',t)f(s , t') + f(s ,t) 
z'>z zz (s'.t'Ms.t)
z'Sz s'Ã¡s
t 'St
= 0.
Remarks.
1) The limits are pathindependent: in particular, in (a), this
limit includes the path where u â–º z along a vertical or
horizontal path.
2) In (b) if s' = s or t' = t, a ,(f)[ = 0, so we can take the
inequalities s' i s, t' S t to be strict. The chosen
definition is simply to preserve symmetry in the limits in (a)
and (b).
3) When we say simply, "f is right continuous," without further
specification, it will always mean in the sense of (a).
4) If f is right continuous, then f is incrementally right
continuous. To see this, note that lAzzfI â€œ f(s',t')
f(s',t)f(s,t') + f(s,t)  = f(s',t')f(s,t) + f(s,t)f(s',t)
f (s, t') + f (s ,t)  (adding and subtracting f(s,t))
S Jf (s ' ,t' )f (s ,t) l + f (s ' ,t )f (s ,t) ] + f (s ,t ')f (s ,t) .
Then f right continous implies each of the three terms on the
23
right tends to zero as (s',t') * (s,t), hence Azz,f >â– 0,
which is (b).
Unfortunately, we do not have the converse implication in
general. To see this, we give the following example, which we shall
refer to later in pointing out further weaknesses of using increments
alone.
Example 2.1.2. Let g be any Evalued function defined on [0,â€œ).
o
For (s,t) e R+, we then put f(s,t) = g(t). Then, for any
(s ,t) 5 (s ', t') , we have
f (s ' ,t ')f (s ' ,t )f (s ,t ') + f (s ,t) [ = g(t')g(t)g(t')+g(t)
= 0.
The function f is then evidently incrementally right continuous for
any f so defined. If we take, however, g to be a function which is
not right continuous, we have
lira If(s',t')f(s ,t) I = lira g(t')g(t)I * 0,
(s',t')*(s,t) t'+t
(s',t ')2(s,t)
hence f is not right continuous. Later on, we shall establish some
additional conditions on f sufficient to have (b) => (a).
Another basic notion for oneparameter functions with regard to
Stieltjes measures is that of increasing function, as we reduce
functions of finite variation to this case via the Jordan
decomposition. Again, we have two definitions, the first the natural
extension of the onevariable definition (order sense), the second
more closely related to measure theory: namely, a condition
sufficient to generate a positive measure.
p
Definition 2,1.3. Let f: R+ > R be a (realvalued) function.
a) We say f is Increasing (in the order sense) if
z < z' => f(z) S f(z').
b) We say f is incrementally increasing if A ,(f) Â£ 0 for
â€”â– â€” â€”â€™zz
all z < z'.
The scalarvalued functions we shall typically consider are defined
using the variation of vectorvalued functions: these (as we shall
see later) are increasing in both senses. In general, however, the
two notions are distinctâ€”neither implies the other, as the following
two examples show. In these, we focus our attention on the unit
square [(0,0) ,(1,1)] for simplicity, but we can extend them (by
constants, say) to give a perfectly good counterexample defined on all
of R2.
+
Examples 2.1.11.
i) We define here a function satisfying definition 2.1.3(a) but
not (b). The particular function we shall give is defined on the unit
square; we could extend it arbitrarily outside [(0,0),(1,1)], but we
shall not give an explicit extensionthe square is sufficient to
indicate how things can go wrong.
The idea consists of writing A ,f = f (s ' ,t' )f (s ' ,t )f (s ,t' ) + f (s ,t)
zz
as f (s ',t')f (s' ,t) â€” [f (s ,t')f (s,t) ], so if the second difference is
larger than the first, the increment will be negative even if f is
incresing in the sense of 2.1.3(a). Accordingly, for (s,t) in the
25
unit square, we define
f (s ,t) = t + s(1  t).
Each onedimensional path, for fixed t, is a straight line connecting
the points (0,t,t) and (1,t,1). Thus, for t < t', the slope of the
section f(*,t) is greater than that of f(,t'), and so the second
difference is larger than the first, so < 0 for ary z < z' in
the unit square. The following computations bear this out:
1) For (0,0) Â£ (s, t) Â£ (s',t'> Â£ (1,1), f(s',t')f(s,t) 4 0. In
fact,
f (s' ,t')f(s,t) * t' + s'(1t')(t + s( 1t))
= t '*s 's't 'ts+st
= (t't)+(s'~s)+St3't'
= (t 't) + (S 's)+sts't+S'tS't'
=â– (t't)+(s's)t(s's)s'(t't)
= (1s') (t't)+( 1t) (s's)
> 0,
since each of the four factors is nonnegative.
2) Denoting z  (s,t), z' = (a ' ,t'), (0,0) Â£ (3,t) Â£ (s',t')
Â£ (1,1),
A ,f = f(s',t')f(s',t)f(s,t ') + f(s ,t )
zz
= t'+s'(1t')(t+s'(1t))(t'+s(1t'))+t+s(1t)
= t ' + S'(1t')t3'(1t)t's(1t')+t + s(1t)
26
= s'[(1t')(1t)]s[(1t')(lt)]
= (s's)(tt') Â¿ 0.
ii) If we set g(s,t) = f(s,t), we get a function g satisfying
Definition 2.1.3(b), but not (a):
g(s',t')g(s,t) = f(s',t')(f(s,t))
 [f(s' ,t')f(s,t)]
Â£ 0,
and similarly A ,g  (A ,f) S 0. We could even create a
zz zz
nonnegative g (g = 1f) with these properties.
As can be seen, these two definitions of increasing are not
nearly so closely related as the definitions of right continuity we
have given. Later, however, we shall give sufficient conditions for a
function f of two variables to have a "Jordan decomposition"
f  f f , where f and f are increasing in both senses of the word.
2.2 The Variation of a Function of Two Variables
In this section we define the variation Varr ,,(f) of a
[z,z ]
2 2
function f: R( E on a rectangle [z,z'] (closed) in R+ and establish
some of its properties. Throughout this section, by "rectangle" we
2
shall mean a closed, bounded rectangle in R+ (but everything goes
2
equally well for such rectangles in R ), unless otherwise specified.
Definition 2.2.1. Let R = [(s ,t), (s ', t') ] be a closed, bounded
2
rectangle un R .
27
a) A partition P of R is a family of rectangles (R.). ,, J
* J JeJ
finite, satisfying the following:
0 0
1) for j,j'eJ, j * j', Rj Pi Rj , = 0 (i.e., any two distinct
rectangles in (R ) are either disjoint or intersect only
on their boundaries)
ii) R  {J R .
JeJ J
(This is a straightforward extension of the notion of
partition of an interval [a,b]CI R).
b) Let P = (R ) , Q  (R ) be two partitions of R. We say
 j jeJ l iel
that Q is a refinement of Â£ if, for each R^ e P_, there exists
a family of rectangles from Q forming a partition of Rj .
(Note: It is evident from the definitions that for j * j'.
The two families from Q forming partitions of R and R , must
be disjoint.)
We show next that any two partitions of a given rectangle R have a
common refinement, as Is the case in one dimension. The main step in
this, and a result we shall use again in its own right, is the
f ollowing:
2
Lemma 2.2.2. Let R  t (s ,t) , (s', t') ] be a rectangle in R , and
P = (R.) be a partition of R. Then there exist partitions
â€” J JtJ
o: s = < s, < ... < s = s' of [s,s'] and r: t = tn < t. <
0 1m u i
... < t = t' of [ t, t' ] such that the family Q of rectangles of the
n
form [(s,t),(s .,t ,)], OSpCm, 0Sq
p q P+1 q + 1 
Remark. A partition of R constructed from partitions o,t of [s,s']
and [t.t'l, respectively, as Q is above Is called a grid on R. We
28
often use the notation oxt to denote the set Q of rectangles as
defined above as well as the vertices of these rectangles. There is
rarely any danger of confusion and where there is we shall be more
explicit. We shall use this notation in the proof of the lemma and
afterward.
Proof of Lemma. For each jE
J J J J J
construction of Q is straightforward: we take o to be the set of all
the Sj and s , ordered appropriately, and t to be the set of all
t and tj ,, put in ascending order. We need to show that oxt is a
refinement of _P. Let, then, R. = [ (s ., t .), (s ' ,t') ] be a rectangle in
J Id d d
P_. Let o' be a partition of [Sj.Sj'] obtained by taking the points
from o between and sj (inclusive), and i' a partition of [tj,tJ]
obtained from t in the same manner. Then o'xt' is evidently a
partition of R.. 1
d
Proposition 2.2.3. Let Â£,P/ be two partitions of R. Then P and P'
have a common refinement, i.e., there exists a partition S of R that
is a refinement of both P and P',
Proof. Let Q = oxt be a grid refining Â£ (Lemma 2.2.2), and
Q' = o'xt' be a grid refining P_'. Denote by p the partition of
s,s'] formed by oo' (put in ascending order), Y the partition of
[t,t'] formed by putting tt ' in ascending order. Then S = pxY is a
common refinement of Q and Q', hence S refines P_, and S refines P',
so S = pxY is our common refinement. (In fact, we have proved that
any two partitions have a common grid refinement.) 1
For a given rectangle R, we can define an ordering on the class Pp
of partitions R as follows: we define P Â¿ Q for two partitions P,Q of
29
R if Q is a refinement of _P. Prop. 2.2.3 says, then, that the class
Pâ€ž of partitions of R is directed under this order.
n
We are now ready to define the variation of a function on a
rectangle.
Definition 2. 2. 9. Let R  [z,z'] = [ (s , t) , Cs', t') ] be a closed,
2 2
bounded rectangle in R and let f: R+ + E be a function.
a) For P  (R.), , a partition of R, R, = [(s . ,t . ), (s ' ,t') ], we
J JeJ j J j J J
define
Varr,  * K fI
L â€™ J jÂ£J j
= I f(s',t')f(s',t )f(s ,tj) + f(s ,t )J.
j eJ
b) We define the variation of f on R, denoted Varr_ _,,(f), by
l z ,Z J
Varr , C f) = sup Varr ,,(f;P) Â£ + â€œ.
[z,z ] K [z,z ]
R
Remark. The supremum in part (b) always exists (finite or infinite),
since the map P â€¢> Var^ z,j(f;P) is lnGreasIn8 for t>le Â°rder defined
above on P . To see this, consider a rectangle [z,z'] partitioned
R
into two rectangles R1 and R2> as in Figure 21.
Figure 21 A partition of [z,z']
30
We have
IA ( f ) J  f(s',t')f(s',t)f(s,t') + f(s,t)
 f(s',t')f(s',t)f(so,t')+f(so,t)+f(so,t')
f(s0>t)f(s,t')+f(s,t)
Â£ f(s',t')f(s',t)f(s0,t')+f(s0,t) + f(30,t')f(s0,t)
f (s,t')+f (s,t) ]
 AR f + hR f
We can do a similar calculation (only longer) for any partition of R,
by adding and subtracting values of f at all the additional vertices
of the refinement, and applying the triangle inequality.
In the next result, we give some properties of the variation.
2
Proposition 2.2.5. Let f: R+ + E be a function.
?
i)For any rectangle [z.z'l C R+, Var^ z,j(f)  0<
ii)Varr ,,(f) can be computed using grids, i.e., partitions of
L Z j Z J
the form Q = ctxt.
iii)Additivity: For 0 Â£ s < s' < s", 0 Â£ t < t' < t" we have
Var[(s,t) ,(s",t')](f) Var[(s,t) ,(s',t')](f)
+ Var[(s',t),(s",t')](f)
and similarly
31
Var[(s,t)(s',t")]
Var[ (s , t'), (s', t") ] ^
(See Figure 22.)
H 1
s s'
Figure 22 Additivity of the variation
iv) If R.C R_, Var (f) < VarD (f).
1 2 R1 R2
v) If f is right continuous (order sense) , we can compute
Varf ,.(f) using grids consisting of points with rational
L Z i Z J
coordinates (and hence we can take the supremum along a
sequence of partitions).
Proof.
i) Since Var. ,.(f;P) > 0 for any partition P, we have
[z,z ]  
Var. ,n(f) = sup Var. ,.(f;P) 2 0.
[z,z ] p Lz.z J
ii) If Var. ,,(f) = + ", then for every N>0, there is a
L Z , Z J
partition P such that Var (f;P ) > N. By Lemma 2.2.2, there
N [z,z'] N
32
exists a grid Q refining Â£ ; by the remark following Definition 2.2. Â¡J,
we have Varr , C f; Q) > Varr ..(fjP,,) > N. Thus, if
[z,z] [z,z] N
Varr ,,(f) = + ", for any N>0 there exists a grid Qâ€ž such that
[z,z ] N
Varr .,(f;Q..) > N, i.e., Varr ,n(f) = sup Varr ,,(f;Q).
[z, z J N Lz,zJ p Lz,z J
We,R
Q=oxx
Similarly, if Varr ,,(f) = a<â€, then for every e;>0, there is a
L Z , Z J
partition P such that Varr ,,(f,P ) > ae. Again, taking a grid
â€”e lz,z J e
Q refining P , we have Varr ,,(f;Q ) > Varr , (f; P ) > ae.
Â£ â€”Â£ LZfZj Â£ !_ Z f Z J Â£
e arbitrary => sup Varr ,,(f;Q) i a = Varr ,,(f). The other
qP Lz,z J L z,z J
Qaxx
inequality is evident, so we have Varr ,,(f) = sup Varr ,,(f;Q).
Lz,z J Q p iz,z j
v R
Q=oxt
(Note: From now on, we shallâ€™compute variations using grids.)
iii) We shall prove the first equality; the proof of the second is
completely analogous.
Denote R = [ (s ,t), (s ' ,t') ], = [ (s ' ,t), (s" ,t') ], R = R1 KJ R^.
For any grid Q = qxt on R, we can add the point s' to o to get a
refinement Q' = Q VJ CÂ¡2, where Q1 is a grid on R1, Q2 is a grid
on R^. Then
Var (f;Q) < Var_(f;Q') = Van (f;Q.) + Varâ€ž (f;Q0)
R R Ri 1 r2 Â¿
Â£ sup Var (f;Q,) + sup Var (f;Q )
Q, R1 1 Q2 R2 2
 Var (f) + Var (f).
R1 R2
Taking supremum on the left, we get
33
Var (f) < Var (f) + VarD (f).
H H^
For the other inequality, if Q1,Q2 are any grids on r r
respectively, then Q is a grid on R, and we have
Var (f;Q ) + Var (f;Q)  VarD(f;Q)
R^ 1 n^j Â¿ n
Â£ sup Var (f;Q) = Var (f).
~ R R
Q=oxt
Since this inequality holds for any grid on R , we have
i .e.,
sup Var (f;Q ) + VarD (f;Q_) < Var tf),
H i H Â¿ K
Q^oxt 1 2
Var (f) + Var (f;Q_) < Var (f).
K ^ R ^ 2 R
Similarly, Q being arbitrary, we have on taking supremum for Q .
Var (f) + Var (f) < VarD(f).
Â°1 â€œ
Putting the two together, we have the equality:
Var (f) + Var (f)  Var_(f).
R1 R2 R
iv) Assume R^ * [ (s ,t) , (s',t') ] is contained in
Rj â€œ [ (q ,r) , (q' ,r ') ] . Then we have q Â£ s < s' Â£ q', and
r Â£ t < t' Â£ r' (see Figure 23). By the additivity property, we have
Var (f) = Var (f) + Var (f) + Var (f) + Var (f) + Var (f).
R2 Ri R3 RH R5 R6
Since each term on the right is nonnegative, we have
Var (f) Â£ Var (f).
K n.
3^
Figure 23 Decomposition of R2
v) Suppose, now, f is right continuous, i.e.,
f(s,t) = lim f(s',t'). We show first that, for any grid Q  oxt
s ' + s
t ' + t
on R e>0, there exists a grid Q' =â– 0'xx' with rational coordinates*
such that
! E Ar f  E K fI! < e.
RjEQ ni RjEQ' j
Let, then, R  [ (s ,t) , (x,y) ], a: s = sâ€ž < s, < ... < s = x be a
u 1 m
partition of [s,x], 1: t = tâ€ž < t, < ... < t â– y be a partition of
0 1 n
[tfy]. We can choose points s' > s_, s' > s' > s , and
u u 1 1 mm
tp > t^, t > tj t' > t 30 that, for each OikSm, 0 S 1 Â£ n,
we have If (s, ,t, )f (s',t,') I < Â¡pâ€” . We take then
1 k 1 k 1 1 4mn
o': s' < s' < ... < s', and x': t' < t' < ... < t'. Then for each
0 1 m 0 1 n
rectangle R^  [(s^,t^) (3k +1 ,t^ + ^ )] Â£ Q, there corresponds a
* Note: Q' is not actually a partition of R, but here f is defined
outside of R so we can use these sums to compute the variation.
The main point of this is to show the variation is the limit of a
sequence, which we shall need later on.
35
rectangle ^ = [ (s ',t^),(s' + 1 ,t' + 1) ] e Q', and we have
I l\ f  IV Ml S ar f  Ar. f
k,l k, 1 k, 1 k, 1
= lf
+f(sk+1'tÃ)+f(3Â¿tÃ+1)_f(sk'tÃ,l
â€˜ ^fCsk+1 ^1 + 1 ^f(sk + l ,tÃ+1 ^ + If(sk+1 ,tl)"f(sk+l I
+ lf(sk'tl+1)_f(3kltÃ+l)l + lf(vti)_fC3Â¿'tÃ)l
< 6 + e + e + e _ e
4mn 4mn 4mn 4mn ran"
Summing up, then, we obtain
 r ar f  i Jar f   e (ar f  ar, f)
RjEQ i RjeQ' j
0
0
k,l
s 1 IK f!  IV fll < E e
k ,1 k, 1
k ,1 k ,1
Now, if Varr ,,(f) = + â€, for every N>0, there is a grid Q with
L Z , Z J N
Varr ,,(f;Qâ€ž) > N + 1. By the above, there exists a grid Q' with
Lz,z J N N
rational coordinates such that Varr ,n(f;Q.',) > N + 7 > N,* hence
L Z , Z J N 2
sup Varr ,,(f;Q) = + Â» Varr , (f). Similarly, if
Q=axx [zâ€™z ] [zâ€™z 1
Q rational
Varr ,,(f) = a < Â°Â°, for every e > 0 there exists Q  oxx such that
Lz,z J E
Varr ,,(f;Q ) > a  and there exists Q' = o'xx' with rational
Lz,z J e 2 E
* Again, is not a partition of R, so this must be interpreted
directly as the sum given in Definition 2.2.4(a).
36
coordinates such that
Var[z>z,](f;V  Var^.^Q'Jl < Â§,
hence
Varr ,, (f; Q') > Varr ,,(f;Q )  Â§ > (a  Â§)  Â§  a  e.
[z,z ] e [z,z J E 2 22
e arbitrary > sup Varr ,,(f,Q) = Varr ,,(f). I
Q=oxt LZ'Z J LZ,Z J
Qratlonal
An important property of the variation in one dimension is that a
function of finite variation f is right continuous if and only if
lira Var. ,,(f) = 0 for all s in the domain of f. Unfortunately,
S'**S [3,S ]
we do not have this equivalence in two dimensions without additional
assumptions about f. We do have one implication, however.
Theorem 2.2.6. Let f: * E be a function with Varâ€ž(f) < Â» for
â– â– â– * â– + R
every bounded rectangle R. If f is right continuous, then for every
2
z, z', u in R with z < z' < u, we have
Varr ,,(f)  lira Varr ,(f).
[z,z ] u++z, [z,u]
(Note: The notation u**z' means u * z', u > z'.)
Proof. We divide the region outside [z,z'] and inside [z,u] into
three parts, labeled R1 , R^, R^ (see Figure 24). The proof consists
R3
R1
N
N,
CM
zs s' p
Figure 24 Decomposition of [z,u]
37
of showing that, as u decreases to z', the variation on each of the
three rectangles R^ , R^, and R^ vanishes. We shall give separate
proofs for R1 and R^, and the proof for R^ is identical to that
for Rj. Note first of all, that for u' such that z' < u' < u, the
corresponding rectangles R, R', R^ satisfy R'C R1 , BjC R,,
R'CR,, so by Proposition 2.2.5(iv), Varn,(f) < VarD (f), etc., and
33 R! R1
so each of the limits lim Var (f), lira Var (f), lim Var (f)
unz' 1 u + + z' 2 u + + z' 3
exists and is nonnegative.
a) Denote z' = (s',t') u = (p,r). We show that
lim Var
p + + s'
r+ + t'
[(s',t') , (p,r) ]
(f)
0.
Assume not: then there exists a>0 such that, for all u>z', we
have Var. , _(f) > a. Let uâ€ž > z', t > 0. Denote u.  (p_,r_).
Lz ,uJ 0 u u u
Since Var. , . > a, there exist partitions oâ€ž: s' = sâ€ž . < sâ€ž , <
[z ,u0J 0 0,0 0,1
sâ€ž < ... < s = p., t.: t' = t  < t , < ... < t.  r_ such
0,2 0,m 0 0 0,0 0,1 0,n 0
that
^ 1 Â®A[(s t ) (s t )]f^ > â€œ
0Â£i
and 0Â£j
ii)
C(so,o,to,o),(so,i ,fco,i) I 2
by right continuity.
(Recall that right continuity implies incremental right continuity.)
We have, then,
VÂ°0XT0
RSCI
Kfl
K fl
VÂ°oXTo
R. C II
E Ar f > <*  Â§â€¢
VÂ°oXTo 8
R6C III
(See Figure 25.) Since each of the three sums is less than the
33
^po,to,i')
Figure 25 Computing variation of R
variation of f on the respective rectangles, we have
Varr. , . , ,,(f) + Varr, , , ,,(f)
[Cs ,t0f1 ).Ca0>1 .h0)] [(s0j1,t ),(p0,t0jl )]
+ Varr , ,, ,, (f) > a  â€” .
[(S0,1 â€™Vi ),(p0â€™r0)]
Denote u1 =â– (sQ 1 ,tQ ^  (p^,r^) > z'. By assumption,
Varr , .(f) > a, so there exists a partition o, : s' = s. . < s, ,
[z ,u] r 1 1,01,1
< ... < s,  pâ€ž, t,: t'  t, â€ž < t, , < ... < t, = r. such that
1,m 11 1,0 1,1 1,n 1
I A f > a and A t f < ? . Then, as
R eo XT 8 0,01,0 1,11,1
p l 1
above, we have that the total variation of f on the three rectangles
comprising [(s ' ,t'), (p1 ,r1 ) ] \ [(s',t'),(s1 1 ,t1 is greater than
a  hence the total variation of f on the rectangles comprising
[(s',t') , (p0,r0) ] \ [(s',t'),(s1 1,t1 > (a  ) + (a  ) =
2  â–
39
Denote  (P2,r2^ * ^S1 t Â»tj By assumption,
Varr , ,(f) > a, etc. Continuing in this manner, we construct a
[z tu2*
sequence uQ, u , u^, ... with > u1+1 > z' for all i such that, for
all i, the total variation of f on the rectangles making up
[(s'.t'l ,(pQ,r0)] \ E(s',t') ,(pi,r1)] is greater than
i
(a  y) + (a  ) â– *... + (a  ^y) = ia  l yr > ia  c,
2 J = 1 J
hence we have Var. , ,(f) > ia  t for all i, i.e.,
Lz ,U0J
Varr .(f) = + Â®, a contradiction of the hypotheses that
[z,u0]
Var (f) < â€¢ on every bounded rectangle. Hence, for any a,<0, we have
n '
0 < lim Varr . . ,, , ,,(f) < a => lim VarD (f) = 0.
p+ + s' [(s ,t ),(p ,r) ] u+ + z. R1
This takes care of R .
b) We show now that lim Var (f) = 0, or, more precisely, that
uâ€œ z 2
lira Varr. , , . . = 0. (See Figure 26). We proceed as
p+ + s' 1(3 jJ
before, by contradiction. Assume there exists a>0 such that
(s,t') z' (p,t'')
z(s,t) (s',t) (p,t)
Figure 26 Computing variation of
40
Var[(s' t) (p t')]^ > â€œ f0râ€™ 311 P > S'' Let> thenâ€™ po > s*â€™ le,:
c>0. There exist partitions a
O'
3 = Sâ€ž â€ž < Sâ€ž . <
0,0 0,1
< s
0,m
V
o1
Vo < Vi < < t0,n
t' such that
VVTo
K fj > a.
Now, consider the rectangle [(s',t),(s0 ^ , t') ]. For each i,
i * 1,2 n, there is r^, s' < ^ < sQ 1 , r^ + 1 < ^ such that for
each i
.iWâ€™Vi
)]
fl <
by right continuity at each (s'.t^ ^ ^).
(rn,t') (s01,t') (pQ,t')
WWWWN
i
i
i
l
t
 V'
V\\\\
! s\\
(s',t) rnr2 (s01,t) (pQ,t)
Figure 27 Breakdown of o^xiq
Subdividing each of the leftmost rectangles of oqxt0 in this
jr creates a refinement P_ of o.xt.., hence E i f[ > a.
Now, we also have that
RyeP â€œy
1 I A[ (s' t ) fr t )]f> < 1 â€œVr < I â€™
i1 Lls â€™V i1 lri â€™Vi â€˜ J i = 1 21 1 Â¿
SO
The rectangles In this sura form a partition of the n rectangles
[(rytMp.t^)], [(r2,t0j1),(p,t0>2)], ... [(vt^Mp.t')]
(shaded rectangles in Figure 27). Thus
n
I Varr, , , . _
i1 [Criâ€™t0,i1) ,(p,t0,i)]
(f)
^ fl " 1 tA [ (s' t ) (r t > a ~
iÂ£p y i=i Lts ,to,ir,iri,to,i;
In particular, then, Var
[(r ,t), (p ,t')] > â€œ " 2 â€¢ Let P1 =
r . By
n
assumption, Var^^, ^ > a> so we can repeat this
* * n â€™
procedure and get another point s' < r' < r such that
n n
Varjj^, ^ > a  j . Continuing in this manner, we can
construct a sequence pQ, p1 , p2> such that
i) Pj > P1 + 1 > s, p^s
ii) Var. , . , ...(f) > a 7TT for alL i*
L(p1 + 1 .t),(Pi,t )] 2i+1
We have, then, by additivity,
Varr
i1
l
â€ž,(f)
j=o
[(PJ+1.t),(Pj
,t')r
i1
i1
l
J=o
(a  ) = ia
2J
 z e
1
j = 0 2J
> ia  e .
i1
Then Var (f) > E Var. > iaE.
R2 j=0 tCPj + 1.t).(Pj,t )]
r\jn
1)2
e arbitrary => Var (f) > ia for all i, hence Var (f) = + ", again a
R2 R2
contradiction. Hence 11m Var.. , , . ,,.(f) i a for any a>0, so
... l(s ,t),(p,t )J
p44S
lim Var., . . . . â€ž,..(f) = 0.
p + + s [Cs ^Mp.t )]
By the same argument, lim Var., ,, . , ,.(f) = 0 (the R.
. [(s,t ),(s ,r) ] 3
case). Putting everything together, we have
lim Var. .(f) = lim [Var. ..(f) + Var (f) + Var (f) + Var (f)]
, . [z,u] ,, . [z,z ] R R R
u + + z u+ + z 1 2 3
 lim Var. ..(f) + lim Var (f) + lim Var (f) + lim Var (f)
u++z' Z,Z u+ + z' 1 u + + z' 2 u + + z+ 3
= Var. ,.(f) +0+0+0) = Var. ..(f),
[z,z ] [z,z ]
which is what was to be proved.
Remarks.
1) As we stated above, the converse is not true. However, if
lim Var. .(f) = Var. ,.(f), then lim Var. , .(f) = 0, and from
. [z,u] [z,z ] u++z' ^Z ,U
u+ + z
Ia. , .fI S Var. , .(f) we get that lim Ia, , .fl = 0, i .e., f is
1 Lz ,u] 1 [z ,u] [z ,u] "
U + + z
incrementally right continuous.
2) Returning to Example 2.1.2, for f as defined there, we have
Var. ..(f)  0 on any rectangle [z,z'], since i fli (f) =
[zâ€™z ] R eP Ra
a â€”
Â£ 0 = 0 for any partition P of [z,z'], so lim Varf ,(f) =
R eP u++z' LZ,UJ
a â€”
Varf However, if f is constructed from a function g that is
L Z , Z J
not right continuous, then neither is f. In fact, for all e>0,
43
f(s+Â£,t+c) = g(t+Â£), SO 11m f(s+Â£,t+E)  lim g (t+ Â£) * g(t) = f(s,t),
Â£+0 Â£+0
so f is not right continuous. In the next section, we shall give
sufficient additional conditions on f for the converse to hold.
2.3 Functions of Two Variables with Finite Variation
As the example in the previous section Cat the end) shows, the
requirement that Var (f) < Â« on bounded rectangles R is by itself
R
insufficient to give all the properties necessary to associate a
Stleltjes measure to it. In order to deduce properties of f from its
variation, we need some extra conditions. It seems natural to require
that each of the onedimensional paths also have finite variation, but
we do not need quite that much. In fact, if Var (f) < Â« on all bounded
R
rectangles R C l, and if the onedimensional path f (â– ,tQ): R+ + E
has finite variation for some t^, then the paths f(*,t) have finite
variation for all t. More precisely, for any s>0, we have
Var[o,s]f(.t) < Var[0>s]f(.
â– V
Var _, â€ž , , , (f).
C( 0,tQ) ,(s,t) ]
(Note: We replace the second term by Varr,â€ž . , , ,,(f) if
L \ U f L J > V S > ^ q / J
t < t.). To see this, let a: 0  s. < s, < ... < s = s be a
0 0 i n
partition of [0,s] (Figure 28): We have, for each i, 0 i i Ã n1,
f(s.+1 ,t)f(Si,t) = f(s1+1 ,t)f(s. ,t)f(s1 + 1 ,t0) + f(3i,t0)
+f(31+i'to)f(si'to)l
< r(sl + 1ft)f(slft)f(3l+1ft0)+fCsl,t0)
+ lf(8l + 1to)"f(8i'to)l
Figure 28 Partition of bounding Varrâ€ž ,f(*,t)
[ 0,3 ]
Summing over the i's, and denoting = [ (Sj ,t ), (s ,t) ], we have
n1 n1
l f Cs
i = 1
ltl,t)f(s.,t) < M4 f + [f(Si+1,t0)f(Si,t0)[)
n1 n1
 1 IAR fl + = 1^3 ,t )f(3 ,t )
i0 i i=0 1 1 u 1 J
Â£ Var[(0,t0),(3,t)]Cf) + Var[0,3]f(',t0)
(or S Var[(0,t),(3,t0)](f) + Var[0,s]fCâ€™â€™t0) if t
Taking supremum over partitions a of [0,s], we get
Var[0>s]f(.,t) < Var[(0(to)j(S(t)](f) â™¦ Var f ^ ]f (â€¢, t Q).
By the same proof (using partitions of [0,t]) we see that if the
onedimensional path fis^,): R+ â– Â» E has finite variation for some
45
3g, then the paths f(s,*) have finite variation for all f, and in fact
(same proof)
Varr ,f (s , â€¢) < Varr, , ...(f)
[0,t] [(sQ,0),(s ,t) ]
Varr_ ,f(s.,
[0,t] 0
Up to now, we have avoided using the phrase "f has finite
variation" because of the weakness of the condition Var (f) < â€œ for R
K
bounded. We shall reserve this term for functions with the additional
conditions described above. We will see that this is enough to give
the additional properties we need to associate useful measures.
2
Definition 2.3.1. Let f: R+ + E be right continuous, with
2
Varâ€ž(f) < Â» for bounded rectangles RCB . We say that f has finite
ft + â€”
varÃ ation if the realvalued function
 f  (s ,t) = f (0,0) +Var[0 g]f(,0)+Var[o fc]f (0, â€¢ )+Var
2
for every (s,t) e R+. We say f has bounded variation
M>0 such that
[(0.0),(s,t)]Cf)
if there exists
< 00
 f  (s , t) < M for all (s,t) e r1
2
The map f: R+ â€¢* R+ is called the variation of f. (Note that we use
the single bars to distinguish it from the norm in E.)
Remarks.
1) Henceforth, the phrase "f has finite variationâ€ will be
2
understood to mean that f(s,t) < 00 for all (s,t) e R+.
2) We extend f by 0 outside the first quadrant to get a
2
function defined on all of R .
46
3) The "jump at zero," f(0,0), will play a role later, similar
to that of the jump at zero in the theory of oneparameter
processes. When we associate measures with f, we shall need this
term to get some compatibility between these measures and those
associated with f,
4) The function f is increasing in both senses of Definition
2.1.3. First of all, if z = (3,t), z' = (s',t'), and z < z', then
f(s',t')  f(s,t) = 0 if (s',t') lies outside the first quadrant;
f(s',t')  f(s,t)  f(s',t') > 0 if z' > 0, z outside R^. If
OS z < z', then
 f  Cs ' ,t')   f  (s ,t) = f (0,0)  + Var[Q g,]f(.,0)
+ Var .,f(0,) + Varr , , _.,(f)
[0,t ] [(0,0),(s ,t )]
lf(0,0)  â™¦ Var[0>s]f(,0) â™¦ Var[0>t]f(0..)
Var[(0,0),(s,t)](f)]
= Varr ,,f(,0) + Varr . â€žf(0,
Ls,3 J Lt,t J
+ (Var[(0,0),(s',t')](f) VarCC0,0) ,(s,t)](f))
> 0.
As for the other sense, we have Ar ,,f = 0 if z' lies outside the
Lz,z J 1 1
first quadrant. We then deal with the case where 0 S z'.
If z is in the third quadrant, i.e., if s<0, t<0, then Ar ,  F 
L z , z J
= f(s',t')  f(s',t)  f(s,t') + f(s,t) =  f  (3' , t ') Â£ 0.
If z Is In the fourth quadrant, 1 .e., if s>0, t<0, then
47
A[z,z']
(f)
f(S',t')
 0
 f(3,
t') + 0
lf(0,0) â™¦
Var[0,s']f(
,0) â™¦ Var[o>t,]f(0,)
+ Var
[(0,0)
,(s'
,t')](0
 ( [f (0,0)  â™¦ Var[0>s]f(
+ Var
[0,t']
f (0,
â– ) + Var
[(0,0) ,(s,t')](f>)
Varr #,f(
[s,s ]
.0)
+ Var[(3,0),(s',t')](f)  Â°â€˜
Similarly, if z is in the second quadrant, I.e., if s<0, t>0, then
A[z,z'](f) = Var[t,t']f(Â°â€™Â° + Var[(0,t),(s',t')](f) S0 Lastlyâ€™ lf
OS z < z', then we have
A[z z'](f) â€˜ _ f(s',t)  f (s.t') + f(s,t)
Jf(0,0) + Varj. Q Â»]f (â€¢ ,0)  Var[0 ^ftO,)
+ Var[(0,0),(s',t')](f) ' (lf(0â€™0)! + Var[0,s']f(*0)
+ Var[0,t]f(0,) + Var[(0,0) ,(s',t)](f))
 (f(0,0) â™¦ Var[0(S]f(.,0) â™¦ Var[0>t .]f (0, â€¢)
* Var[(0,0),(s,t')](f)) + + Var[0,s/(â€™0)
+ Var[0,t]f(0,0 + Var[(0,0),(s,t')](f)
(Var[(0,0) ,(s',t')](f) " Var[(0,0) ,(s',t)](f))
(Var[(0,0),(s,t')](f) ' Var[(0,0) ,(s,t)](f))
Â¡J8
Var[(o,t),(s',t')](f) " Var[(o,t),(3,t')](f)
Var[(s,t),(s',t')](f) Â£ Â°
This definition allows us to recover many results analogous to
those of functions of one variable, as we show in the next few
theorems.
Theorem 2.3.2. Let f: + E have finite variation f. Then f Is
right continuous if and only if f is right continuous.
Proof. Assume, first, that f is right continuous. We write, for
(s,t) > (0,0),
f(s,t)
â– f(0â€™0) + Var[0,s]f(â€™0) + 7ar[0,t]i'<0â€™ â€˜ 1 + var[(0,0),(s,t)](f)
The first term is constant; to show that f(s,t) = lim f(s',t"), it
s '+s
t ' + t
suffices to show that
15 aâ€œâ„¢s Var[0,s']f(0) â€œ Var[0,s]f(â€™0)
ii) ^lim^ Var[0^t .]f(0,â€¢)  Var[0_t]f(0,â€¢)
ill) lim Var ni = Varr^n oi t *\(f)
(s',t')++(s,t) [(00).(s .t )] [(0,0),]
We proved (iii) in Theorem 2.2.6 (taking z = (0,0), z' = (s,t),
u = (s',t')). As for the other two, f right continuous implies
f(.,0), f(0,â€¢) right continuous: in fact, taking t' = t = 0, we have
H9
lim f(s',t') = 11m f(s',0) = f(s,0)
(s',t')+(s,t) s' + + s
(s'.t'Ms.t)
(Recall that the definition of right continuity allows us to take
limits along vertical or horizontal paths as well, unlike left
limits.) Hence f(*,0) is right continuous, so the variation is right
continuous, i.e., Var f(,0) = lim Var f(,0). Similarly,
LU.SJ s'**s L0,s J
taking s' = s = 0, we have
f(0,t) = f (s, t)  lim f(s',t') = lim f (0, t'),
(s',t') + (s,t) t' + + t
(s',t'Ms,t)
so f(0,â€¢) is right continuous. The variation is then right
continuous, so Var ,f(0,) = lim Varf ^(0,). Then each of
the terms of f is right continuous; hence f is right continuous on
Conversely, assume f is right continuous at each point
2
(s,t) e R+. Taking t  0, and letting s'+ + s along the path t = 0, we
have
Var[0,s']f('â€™0) " Var[0,s]f(',0) â€œ lfI f(s,0).
In fact,
50
IfI (s ' ,0)  f(3,0)
+ Var[0,3']f(â€™0) + Var*[0.0]f(00
+ Var[(0,0),(3',0)]f  (lf(0'Â°l + Var[0,s]f(0)
Var[o,o]f(0â€™Â° + Va"[(0,0),(s,0)]f)
â– Var[0,s']f(>0)  Var[0,s]f(â€™0)
Then Varrâ€ž _f(â€¢,0) = lim Varrâ€ž ,.f(,0) since Ifl is right
[0,s] s,+ + s [0,8 ]
continuous, i.e., the variation of f(,0) is right continuous; hence
f( ,0) itself is right continuous. A similar computation taking s = 0
shows that f(0,*) is right continuous. Then, writing
Var
[(0,0) , (s,t)]
(f)  f(s,t)
If(0,0)  Var[0j3]f(,0)
 Var
[0,t]
f(0,),
we see that each term of f is right continuous.
Now, let Cs'.t') 2 (s,t) , (s',t') * (s,t). We have
f(s',t')  f(s,t)  f(s',t')  f(s',t) + f(s',t) 
S f(s',t')  f(s",t)  + f(s",t)
f(s,t)[
 f(s,t) .
We note now the following inequalities (cf. Figure 29):
1) f(s',t')  f(s',t) = f(s',t')  f(s',t)  f(0,t') + f(0,t)
+ f(o.t')  f(o,t) I
S f(s',t')  f(s',t)  f(0,t') + f(0,t) I
+ f(O.t')  f(0,t)
51
' lAC(0,C).(s'.t')]fl + I*10'1"* ~ f(0â€™t,l
Â£ Var[to,t) ,(s',t')](f) + lf(0â€™t') *
2) f(s',t)  r (3, t)  = f(s',t)  f(s,t)  f(s',0) + f(3,0)
+ f(s',0)  f(s,0) 
S f(s',t)  f(3,t)  f(s',0) + f ( S , 0 ) 
+ f(s',0)  f(3,0)
â€˜ i[(3.0),(3',t)](f)l * " f(3â€™0)l
Â£ Var[(3,0)t(3'(t)](f) + lf(3'0)  f(3>0)l
Putting everything together, we have
f(3',t')  f <3, t) J < f(3',t')  f (s',t) J â™¦ f(s',t)  f(s,t)
Â£ Var[(0,t) ,(3',t')](f> + If(0,t,) â€œ f(0,t)l
+ VaP[(3,0),(3'.t)](f) + lf(S'0)  f(Sâ€™0)l
â– CVar'[(0,0),(s',t')]if) ' Var[(0,0),(3,t)](f)]
+ fCo.t')  fCO,t) ] â™¦ f(3',0)  f(3,0)0
Ccf. Figure 29). As we showed above, each of the three terms on the
7
/
/
/
s s'
Figure 29 Rectangles used in (1) and (2)
52
right tends to zero as (s',t') decreases to (s,t), so we have
lira f (s ', t')  f (s, t)  Â» 0,
(s',t')*(s,t)
I.e., f is right continuous. I
Remark. We still have the same result If we extend f, f by zero
outside Rr".
+
The next result concerns the existence of "onesided" limits and
"limits at infinity."
Theorem 2.3.3. a) let f: R^ * E be a function with finite variation
f. Then each of the following limits exists* at each z = (s,t) e R^
1)
2)
3)
4)
f (s+,t+)
 lim
3 '++3
t '* + t
f(s'.t')
f(s_,t+)
Â» lim
3 'its
t '**t
f(s'.t')
f(s_,t_)
= lim
S 'MS
t 'ttt
f(s',t ')
f(3+,t_)
= lim
s 't*s
t 'ttt
f(s'.t')
Moreover, if f has bounded variation (i.e., if there exists M>0 such
that f(s,t) < M for all (s,t) e R^), then each of the following
* Of course, on the axes, not all these limits make sense. It will
be understood that at each point we take limits from quadrants
where f is defined.
53
"limits of infinity" exist:
1') f(s+,Â°Â°) = lim f(s\t')
3 '44 s
t ' + Â«
2') f(s_,Â°Â°) = lim f(s',t')
S ts
t
3') f(â€œ,t+) = lim f(s'.t')
t ' + + t
S 'too
^') f(â€,t_) = lim f(s',t'), and especially
t 't + t
S 'too
5') f(oo) = lim f(s'.t') exists,
s', t 'tÂ«>
b) If, moreover, f is right continuous, then the onesided limits
along the vertical and horizontal paths f(s,), f(*t) are equal to the
following.
i)lim r(*,t) = lim f(s,) = f(s+,t+) (right limits)
s' + *s t' + + t
ii)lim r(*,t)  f(s,t+), and
s 'tts
lim f(,t) = f(oÂ»,t + ) if f is bounded.
iii)lim f(s,) * f(s+,t), and
t 'm
lim f(s,) = f(s+,â€œ) if f is bounded,
t
O
Remarks. 1) Here f is defined on R : if we wish to use an f defined
+
also on the "boundary at infinity," the above limits will be denoted
with the symbol Â» in place of Â«.
2) In general, a function of finite variation can have eight
different limits at a point: the four "quadrantal" limits from part
(a) of the statement, plus the four onesided limits along the
vertical and horizontal paths. Part (b) of the statements says that
if f is right continuous, the onesided limits can be incorporated
into the quadrantal limits, so there are only four distinct limits at
a point (s,t) (at most). The first part of (b) says that both right
limits along the vertical and horizontal paths are equal to limit (1),
the second part says the left limit along the horizontal path is equal
to limit (2), and the third part says the remaining left limit is
equal to limit (il) , giving the division of the plane shown in Figure
210. There are analogous considerations for the "limits at
infinity."
J
(s,t) (1)
(2) (s,t)
(s t)
(3)
(4)
Figure 210 The four "quadrants
55
Proof, a) Assume first that f has finite variation f.
The proofs of limits (Ot'O are similar; we treat (1) first. We
shall show that for any sequence (s ,t ) with s ++s, t **t, the
n n n n
sequence lf(s ,t )} â€ž is Cauchy in E. We shall do this by denial:
n n neN
Let (s ,t ) be a sequence as above and assume that f(s ,t )} â€ž is
n n n n neN
not Cauchywe shall reach a contradiction.
Since f(s ,t ) is not Cauchy, there exists e_>0 and a subsequence
n n 0
(n, ), â€ž such that, for all k, we have
k keN
fCs ,t )  f(s ,t ) > En.
â– n, , n, , n, n, â– 0
k+1 k+1 k k
We will henceforth denote this sequence by lf(s ,t )} so we have
n n neN
the inequality for all n.
For j given, consider the subdivisions si > sR > ... > s^
> Sj + 1  s of [ s, s ^ ] and t1 > t2 > ... > tj > tj + 1  t of [t,^].
For 1 = 1,2 J1 denote Rj = [ (s ,0), (s , tJ + 1) ] and
R' = [(0,tJ+1),(s ,t )] (Figure 211). For each i, we have
'I  I^VVl5 â€˜ f(s1+1â€™ti+1) â€˜ f(V0) + f(si + 1â€™0)â€™ so
hR f 2 lf(siti^i> " ^VrS+i^ â€˜ !f(si0) ' f(si+i0)l >
f(V0) ' f(Sl + 1 '0) I + lAR fl  I^VVl5  + + ^ haVS
similarly
AR.f[  f(s1.ti)  ffSj.t^)
f (O.t^ + f (0,tj +1) 
> If(sj,11)  f(sIftl + 1)  Ifto.tj)  f(0,t1 + 1
56
Figure 211 Partition of [(0,0) ,(s,t) ]
> f(0,t.)  f(0,t. + 1)l * 1 ARrf 1 Â£ f(si,t1)  f
the two together we have, for each i
lf
= f(3.,t.)  f(Sj,t.tl) + f(s.,t.+1) ' f<3l*1'tit1)l
< !f(s.,ti)  f (si ,t.+1 )  â™¦ f(a1,t1 + 1)  f(s1 + 1,t. + 1)l
 ]ar r + iR.f  lf(sÂ¡,o)  f(3.+1,o) + rco.t1)  f(o,t1+,).
1 Ã
Now, upon summing over the iâ€™s, we have
57
 f(Si+1*tl + l)l
j1
* s
i = 1 i i 1 11
+ Ifto.tj)  f(0,t.+1)[)
J1 j1
 Ã fl + hR.f) + I f(s 0)  f(s 0)
i1 i i i = 1
J1
+ E f(0.ti)  f(0,t1+1)
 Var. , . ..(f) + Var f (.0) + Var
[(0,0) ,(s1 ,t1 )] [s .Sj] [t ,t
(since the union of the R^, R' is contained in [(0,
 Varr/ n an i + , t (f) + Var. .f(,0) + Va r.â€ž
[ ( 0,0),(s1,t1) ] [Â°j Â»1 [0,t1
s )r(Sl,t1>.
Ji
We end up with l ffÃs^t^  f(si + i * I   f  (. t1) â€¢
each term on the left is greater than e^, so we have
J1 .
I IfCSj.tj)  f(si + 1 .t1 + 1)  > (jDe0, hence  f) (s1,t1)
The left hand side does not depend on j , so letting j Â» Â®
f(s ,t )  + Â», a contradiction on the assumption that
,f(0,*).
1 J
0),(sl,t1)])
jf (0, â€¢)
Now,
, we obtain
IfI < â€¢
58
Thus, for any sequence (s ,tn) decreasing to (s,t), the sequence
â– f f (s ,t )] is Cauchy in E complete; hence lim f(s ,t ) exists. We
n n ncN n n
n
show now that we get the same limit for any sequence decreasing to
(s,t).
Consider two sequences (s ,t ) and (s',t'), both decreasing to
n n n n
(s,t). We construct a new sequence (p ,r ) as follows.
We set (p ,r ) = (s^tq, Cp2>r2)  the first term of (s^,t^)
smaller than (p^r^, (p^,r^) â€œ the first term of (s^.t^) after
(s .tq smaller than (p2>r2), etc. The sequence (pn,rn) then
decreases and converges to (s,t) since both the even and odd
numbered terms do. Then L =â– lim f(p ,r ) exists from above. Looking
n n
n
at the oddnumbered terms, we have L = lim f(P2k+1,r2k+1^â€˜ Sut the
k
oddnumbered terms form a subsequence of (s^,^), and we know
fCs ,t ) converges, so L = lim f(s ,t ) as well. Similarly, since
n n n n
n
the evennumbered terms form a subsequence of (s',t'), we obtain
n n
L  lim f(s',t') as well. The limit is then independent of the
n n
particular sequence, so lim f(s',t') exists, and it is
(s',t')Ws,t)
this limit we denote by f(s ,t+).
The proofs of (2)(Ji) are similar, and we will omit some
computational details where they are identical to the ones for (1).
Proof of (2). We consider a sequence (s ,t ) * (s,t), with s ms,
n n n
t ++t, and show that the sequence {f(s ,t )} â€ž is Cauchy in E, which
n 1 n n JneN
we again do by denial.
59
As in the proof of (1), we extract an > 0, and (s ,tn) such
that (s^.t^) Â» (s,t), and for each n we have sn+1 > sn> tfi+1 < tn,
and f(s , ,t ,)  f(s ,t )] > c_. As in (1), for given J, consider
â– n+1 n+1 n n â– 0
the subdivisions s1 < s2 < â€¢â€¢â€¢ < sj < s^ + 1 = s of [s1 ,s] and
t1 > t2 > ... > tj > tj + 1  t of [t,^]. For i = 1,2,...,j1, denote
R1 = [ (s ,0) ,(sl + 1 ,t Â±) ], R' = [( 0,tl + 1 ) , (s . +1 ,t.) ]. (See Figure 212.)
We have, for each i (similar to before):
IARlf â– lf
>  f (SÂ± ,tt1) I  f(sl + 1,0)  f (s . ,0) ,
hence
1
â– v
V
v.
2
\
\
\
i
Ri'
\
V
i+l
\
V
N
R.
l
s
1 s
2"Â£
. s. 1 
i i + l
s
Figure 212 Partition of [ ( 0, 0) , (s ,t) ]
60
f(s.+1>0)  f(s ,0) + JAR f i f(s ,t )  f(3 ,t )
i
I\fl â– lr(ai+i'ti)  f'ViVi1  f(0V +
2 fCai+1.ti)  f(s1+,.t1+1)  f(0.t.)  f(o,ti+i),
so
 f(0,t1 + l)l â™¦ AR.f 2 f(3l + 1>t )  f(s ,t ).
i
Here we diverge from the proof of (1), since the rectangles R , R'
overlap (see Figure 212). From the first inequality we have, upon
summing over i,
J1
Z f(s t )
i = 1
j1
f
j1 j1
E Ar f + E f(a1 + 1,0)  fCs1.0) 
i=1 i i1
< Varr. ,n(f) + Var_ _,f(,0)
[(0,0),(s,t )] [s ,s]
as in the proof of (1)
Â£ f(s,t ) (as in the proof of (1)).
Also, by a similar computation, we have
61
j1
Â£ If(s ,,t.)  f(s. ,,t, ,)
i_1 * i+1 i 1 + 1 i + 1 "
j1 j1
< z AR,r  z f(o,t.)  f(0,tI+1)
i = 1 i i = 1
Â£ VarU0.0)As,t^r) +
Â£  f  (s, c 1 >.
Putting the two together, we have (as in (1)):
j1
I If(3i,t1)  f(s1+1,ti+1>
1 = 1
Vf(Si.ti)f(S1 + i.t1) + f(3i + i.ti)f(3l + i,Vi)l
<  f(si + 1,t1) â™¦  f(s1 + 1,t1+1)
i1
11
fI(s,t1) + IfI(s,t1)
2f(a,t ).
Again, each term on the left hand side is greater than e^, so we get
j1
(J1)e < Â£ f(s ,t )
1 = 1
f(si+1â€™ti*1)l S 2lfl(3â€™t1)
Letting j â– Â» Â»,
we get f(s,t ) = + â€, a contradiction as in (1); hence the sequence
f(s ,t )] is Cauchy in E complete, so lira f (s ,t ) exists. The
n n n n
n
remainder of the proof is exactly the same as that of (1).
62
The proofs of the last two are the same as those of the first
two: to prove (3), we use the same method as that of (1) to get
j1
E Â¡f^Sj.tj)  f (s1 + 1 ,ti + 1 )  Â£ I f I (s,t) (instead of f(s1,t1>>,
and to prove (A) we use the same method as that of (2) to get
j1
z f(sl,tj>  f(s1 + 1tj + T) I  fICsl,t) (instead of f(s,t1)).
i = 1
This completes the first part of the theorem.
Assume, now, that f has hounded variation, i.e., there exists M
such that f(s,t) < M for all (s,t) e R^. We will prove the
existence of the "limits of infinity" in pretty much the same manner
as the proofs of the other limits: the main difference occurs in
using M instead of a particular value of f to obtain a
contradiction.
Proof of (1') . Let (s ,t ) be a sequence with s + + s. s . < s
n n n n + 1 n
Q
V = for all n, t ++<â– >. We proceed by denial as above. The proof of
this is much the same as that of (A) (and (2)), with a slight
difference: proceeding in the same fashion as in (2), we obtain
j1 j1 j1
I f(s ,t )  f(s ,t ) < E ]A r + I f(s ,0)  f(s ,0)j
11 i1 i i1
Â£ f(Sl,t.)
(see Figure 213). However, the right side now depends on j, so we
must further majorize it by M:
63
Ri
\
N
Ri
S Sj "" Si+1 si """ Â£1
Figure 213 Partition for the "limit of infinity" f(s*,Â»)
J1
1 IftSi.tj)  fÃVrVi * Ifl(3i M
Similarly,
 f(sl + l,ti+1) S  f  (s, .tj) S M.
Now, as before, we have
J1
(JDc0 < Z f(s1>tl)  f(si + 1,ti + 1)
J1 J1
< I Ifis^tp  f (si + 1,t i) I + E lrisi+i ' f(si'ti1)
< M + M  2M,
and we get a contradiction as before. Then f(s ,t ) is Cauchy; hence
n n
lim f(s ,t ) exists, and we prove the limit is the same for any
n
sequence the same way as before.
This illustrates the difference between the proofs of (1)(4) and
those of (1')(5'): We do the same computation for the limits at
infinity, but the value of f turns out to be at a point depending on
j, so we further majorize it by M. For (3') we have
j1
I Ifis^tj)  fSj) Â£ M
j1
iZ1lf(si+1â€™tl) â– f(si+1â€™ti+1)l S lrl(tl'sj5 Â£ Ml
so as before
j1
(j1)Â£0 < i  f(81 + 1,t1+1) Â£ 2M,
and we conclude as above. For (2'), (4'), and (5'), we follow the
same computation as in the proof of (1) and obtain
j1
(j1)e0 Â£ Z IfiSj.y  fC31 + 1,t1 + 1) Â£ IflÃSj.tj) Â£ M
and conclude as in the proof of (1). This completes the proof of (a).
Proof of (b). Assume, now, that f is right continuous (order
sense!). We shall deal with (u) and (ui) first).
65
Ad (1 i). Denote L *= f(s,t+), let e > 0. There exists { > 0 such
that for all (s',t') with s't, ss' < 6, t't < 6, we have
f(s',t')L <  . Let, then, s' < s with ss' < 6: since f is right
continuous, there exists a point (s",t') with s' < s" < s, t' > t,
t't < 6, such that f(s',t)  f(s",t') < ^ . But we also have
f (sâ€™1 ,t')L  < ^ , hence [f(s',t)L Â£ f(s',t)  f(s",t') +
 f (s" , t')L  <â€œâ– *â– â– â€”= e. Thus, lim f(s',t)  L  f(s,t + ).
s 'tts
The proof of the limit at infinity is much the same, except that
instead of ss' < a, there is N such that for all s' > N, the
conditions hold. We then take s' > N, s" > s, and the remainder is
the same.
Ad (ill). The proof of (iii) is the same as that of (ii), with the
roles of s and t being reversed: for e > 0 there exists 6 > 0 such
that for all s' > s, t' < t, etc. The remainder is the same.
Ad (i). This follows immediately from the definition of right
continuity: all three limits are equal to f(s,t). We should remark,
however, that it is the same to define right continuity using the open
quadrant: the limits along the horizontal and vertical paths are then
the same as the "quadrantal" limit. In fact, for e > 0, choose S > 0
so that for s' > s, t' > t, s's < 6, t't < 6, f (s,t)f (s ',t')  <  .
Then for any s' > s with s's > 6, there is a similar " ^ 
neighborhood" for the point (s',t). Pick any point in the
intersection of these " ^ neighborhoods," and apply the triangle
inequality as before. B
Our next result concerns the existence of a "Jordan
decomposition" for functions of two variables with finite variation:
66
in two variables, we have two distinct definitions of "increasing,
but our decomposition satisfies both.
2
Proposition 2.3.1*. Let f: R+ + R have finite variation (again, in
the sense of Definition 2.3.1). Then we can write f = f  f^, where
f and f are increasing in both senses of Definition 2.1.3, namely
a) for (s,t) Â£ (s',t') we have f^s.t) Â£ f^s'.t') and
f2(s,t) Â£ f2(s',t')
and
b) for (s,t) < (s',t'), A . ,.,(f ) Â£ 0 and
L(s,t),(s ,t )J 1
Ar, f.' , ^ 0 â€¢
L(s,t),(s ,t )J 2
Proof. For (s,t) c R^, set f^s.t) = f(s,t), f^fs.t) 
f (s,t)  f(s,t) = f(s,t)  f(s,t). In remark (U) following the
definition of f (Definition 2.1.3), we showed that f is increasing
in both senses. We then have only to deal with f .
a) Let (s,t) Â£ (s',t'). We have
f (s',t')  f2(s,t) =  f  (s'.t')  f(s',t')  (  f  (s, t)  f (s, t))
=â– (  f  (s ', t')   f  (s, t))  (f(s'.t')  f (s, t)).
We shall show f(s'.t')  f(s,t) Â£ f(s',t')  f(s,t). Denote by
R1 the rectangle [(0,t),(s',t') ], by R2 the rectangle [(s,0),(s',t)]
(Figure 21H). We have
f(s',t')  f(s,t)
Â£ f(s',t')  f (s, t) 
67
f(s',t')  f(s',t) + f(s',t)  f(s,t)
Â£ ff(s',t')  f(s',t)[ + If(s',t)  f(3,t)
= A f + f(0,t')  f(0,t) + a f + f(s',0)  f(s,0)
1 2
S  A f[ â™¦ I f (0, t')  f (0,t)  * Â¡A fj + f (s ' ,0)  f (s , 0) Ã
< Var (f) + Varr ^fiO,) + VarD (f) + Varr ,,f(*,0)
R1 [t,t ] R2 [s,s ]
= (VaP[(0,0) ,(s',t')](f) " Var[(0>0)>(sIt)]
+ Varr ,,f (0, â€¢) + Varr =,,f (â€¢ , 0)
LtjtJ Ls,s J
f (s',t')  I f I (s,t)
(cf. Remark 4 following Definition 2.3.1).
R1
V
\
\
\
\
\
\
(s,t)
R2
(s'.O
Figure 21U Bounding the difference f(s',t')  f(s,t)
Then f(s',t')  f(s,t) Â£ f(s',t')  f(s,t), hence
68
f2(s',t')  f2(s,t) = (  f  (s ',t')  f(s,t))  (f(s'.t')  f(s,t)) > 0,
i.e., f2(s, t) < f (s',t#).
b) Let (s,t) < (s',t'): we have
A[(s,t) ,(s',t')](f2) f2(s â€™*â– 5 " f2(s,,t) â€˜ f2(sâ€™t') + f2(s,t)
=  f  Cs t')  f(s',t')  (]f(s',t)  f(s',t))
 (f(s,t')  f(s,t')) + f(s,t)  f(3,t)
 (f[(st')  f(s t)  fC s,t') + f(s,t))
 (f(s'.t')  f(s',t)  f (S, t ') * f(3,t))
= A[(s,t),(s',t')](lfl 5 " A[(s,t) ,(s',t')]Cf)
= Var,[(s,t) ,(3',t')](f) " A[(3,t),(s',t')](f)
Â£ 0.
Thus f^ is increasing In both senses, and the proof is complete. I
Remark. Both 2.3.3 and 2.3.^ hold with f and f extended by zero
outside the first quadrant.
CHAPTER III
STIELTJES MEASURES ON THE PLANE
In this chapter we extend the classical correspondence between
functions of finite variation on the real line and Stieltjes measures
on the real line to the case of functions and Stieltjes measu"es on
3.1 Measures Associated With Functions
2
Given a function f: R * E with finite variation on bounded
o
rectangles, right continuous (in the order sense!) on R , we can
associate a unique measure with finite variation. The statement and
proof we give are due essentially to Radu [16]. The statement is a
little more general than we really need, but no further difficulties
are encountered by this; we also use rightlimits instead of Radus's
leftlimits, but this is just a matter of choice. The term "bounded
variation" in the statement refers to the variation of f on rectangles
as in Definition 2.2.11; as we have seen, this is weaker than the
requirement that f be bounded.
2
Theorem 3.1.1 (Radu). If the function f: R + E is of bounded
variation on R^ and if the right limit f(s+,t + ) (cf. Theorem J/.3.3 for
2
definition) exists at each point (s,t) of R , then there exists a
2
Stieltjes measure m on R with values in E, uniquely determined,
with finite variation, and such that for all rectangles
69
70
R * ((s, t) , (st') ] we have
m(R) = f(s'+,t' + )  f(s'+,t+)  f(s+,t'+) + f(s+,t+)
" Yf*>
Proof. We give the proof in several steps.
. ?
1) Let 6 be the family of rectangles of R of the form
(z,z'], z
measure theory; In fact, for any sets S,T with semirings of subsets
5, T, respectively, the family of sets of the form AxB with
A Â£ S, B e T is a semiring of subsets of SxT. Here S = T  R, with
S,T the semirings of halfopen Intervals.
We define a set function o: 5 â– * E by
o(R) = Aâ€ž(f ) = f(s'+,t'+)  f(s'+,t+)  f(s+,t'+) + f(s+,t+)
n +
for R  ((s,t),(s',t')] with (s,t) < (s',t').
2) a is additive on 6. Let R^, e {, disjoint with
R1 U e Now> R r2 a^s0 a rectangle iff they "match up" on
one side (see Figure 31).
I
I
I
I
1"
R2
1
1
1
t
l
xâ€”i 1
Â« ,
1
s
s'
Figure 31 Additivity on 6
Denote
R.] = (s,s'] x (t,t']
Rj = (s,3'] x (t',t"]
(Figure 31). (The proof is the same if is of the form
(s',s"] x (t,t'].) We have
m(R UR2)  f(s"+,t"+)  f(s+,t"+)  f(s'+,t + ) + f(s+,t + )
 tf(s'+,t"+)  f(s+,t"+)  f(s'+,t"+) + f(s+,t'+)]
+ [f(s'+,t' + )  f(s+,t'+)  f(s'+,t + ) + f(s+,t+)]
(we added and subtracted f(s'+,t'+) and f(s+,t'+))
 m(R ) + m(R ).
3) 0 has finite variation on 6. We prove this by contradiction
suppose there exists a rectangle J t 5 such that o(J) = + Â» ()o
denotes the variation of 0). Then, for each a>0, there exists a
finite family (J ), h * 1,2,...n of disjoint rectangles from 6,
J, C J for all h, such that
h
n n
E o(J )  > a, i.e., E [A (f+)  > a.
h=1 h1 n
Denote J.  ((s,_,t.) , (s ',t/) ] for all h. We may, of course, assume
h h h h h
n
that J = IJ (so that \_Jj c 6). Let e > 0. Since f has
n1 h
72
rightlimits everywhere, there exists a number p > 0 common to all the
vertices of all the J, such that
h
If(V'V>  fI < t 
lf(sh+,th+) â– f(sh+p,th+p)I < Ã 
lf(sh+,th+) ' fCVp,th+p)l < t  and
Iris'^V5  f(Sh+P.th+p)  < f for all h.
n
If we denote L = [ (s, +p ,t, +p), (s,' + p, t ' + p) ], then 1 = 1) I is a
h h h h h pâ€”'. h
h= 1
closed, bounded rectangle in R2, and the family P = (i^: h = 1 .. .n}
forms a partition of I according to Definition 2.2.1. Also for each
h, we have
A , (f+)  At (f)
Jh Xh
= f(s^+,t^+)  f(s^+,th+)  f(s
 (f(sÂ¿+p,tÂ¿+p>  f(s'+p,th+p)
=  f(s'+P,t'+P)) 
 (f(s +,t'+)  f(s +p,t' + p))
n h n n
s lf(sh+,th+) â€˜ fCsh+p,th + p) I +
+ lf(sh+,th+) â€˜ f(sh + p,th+p) I
. Â£ E E E
*_ +  + r + T = E
 f(sh+p,tÂ¿+p) + f(sh+P.th+P))
(f(s^+,th+)  f(s^+p,th+p))
+ (f(sh+.th+) â– f(sh+p,th+p))I
lfCsh+,th+) â– f(sh+p,th+p)H
+ If(sh+,th+) ' f(sh+p,th+p^ I
73
In particular, we have
hj CD I > a (r+)  E.
h h
Upon summing over h, we obtain
Var (f;P) = I at (f) > 1 A T (f+)  ne > a  ne.
h h h
Now, denote J  ((p,r) , (p ',r ')] : we can take p<1, and decreasing with
e, so for any e, we ha ve (J I, = ICI[(p,r),(p'+1,r' + 1)]. Denoting
h
this latter rectangle by K, we have K ID I for all I (in general, I
depends on e) , so Var (f) > Var (f) > a  ne. e arbitrary =>
K I
Var (f) > a. Now, the collection J depends on a, but they all have
K n
union equal to J, so we can repeat the above procedure for any a and
keep IC K, Thus, Var (f) > a for any a => Var (f) = + Â», a
K K
contradiction on our assumption of finite variation of f. Then a has
finite variation on 5.
4) a is inner regular on 6, We observe first that, from the
2
fact that the rightlimit of f exists at each point of R , it follows
2 2
that for each z e R , e > 0, there exists z' e R with z
for any u with z < u < z' we have f(u+)  f(z+) Â£ e. In fact, since
f(z+) exists, there exists n>0 such that for any points u,v>z, with
uz
point with zz' < n, z
f( u+)f(z') Â£ ^ . Likewise, letting v**z, we have
f(v)f(z') <  => f(z+)  f(z') <  . Then f(u+)  f(z+) Â£
lf(u+)  f(z') + If(z')  f (z+) I Â£  +   E
7M
Now, let J e Ã, J = ((s,t), (s',t') ] with s0.
There exists a point (p.r) e J such that for any point (h,k) with
(s,t) < (h,k) < (p,r) we have f(h+,k+)  f(s+,t+)  <  ,
f(s+,t'+)  f(h+,t'+) < j , f(s'+,t+)  f(s'+,k+) <  , as in
Figure 32. (We can do this for each by the above, and we use a
common n in choosing our (p,r).)
(s",0
(s',k)
,t>
Figure 32 Approximation of a rectangle from within
Let, then, K = [(p,r),(s',t')] compact. Any rectangle J' from
6 such that KC J'C J must be of the form J'  ((h,k),(s',t')]
with (s,t) < (h,k) < (p,r) (see Figure 32). We have, then,
Jo(J)  o(J')  j A (f +)  A , (f + ) I
J J
= f(s'+,t'+)  f(s+,t'+)  f(s'+,t+) + f(s+,t + )
75
 (f(s'+,t'+)  f(h+,t'+)  f(s'+,k+) + f(h+,k+)
Â£ f(s'+,t'+)  f(s'+,t' + )  + f(h+ ,t'+)  f(3+,t'+)
+ ff(s'+,k+)  f(s'+,t+) + f(h+,k+)  f(s+,t+)
<
e
4
e.
hence o is inner regular on 6. It follows, then, by Proposition 19
[6, p. 314], that o is also inner regular on 6.
5) Let t(6) be the class of subsets H C R2 for which e 5
for any J e 6. Since o is additive on 5, o is additive on t(6)
(standard result from measure theory), hence o is additive
on 5Ct(6). We shall now denote o by V (for clarity in what
follows).
Let o, V be the additive set functions obtained (uniquely) by
extending o and V to the ring C generated by 6. We show next that
n
o has finite variation on C. In fact, let A e C. Then A = [^J A ,
1 = 1
A disjoint, Aj e 6. We have
n _ n
o (A) I   o (.{J A, )  =  I o (A )  Â£ l o(A ) =
i i1 11
n n n _ _ n
= I o(A.) Â£ l V(A ) = I V(A ) = V(U A ) = V(A).
i1 1 i=1 i=1 i1
Then o(A) Â£ V(A) for all A e C, so o Â£ V (since o is the
smallest positive measure dominating o()), hence o has finite
variation.
76
6) Since c is inner regular on 6, o is inner regular on C = R(S)
[6, Corollary to Prop. 7, p. 308], We now show o is regular on C.
This follows immediately from the following proposition [6, p. 306]:
Suppose that the ring C satisfies the following
condition: for every set A e C there exists a set A' e C
such that AC Int(A').
Then a measure m is regular on C if and only if m is
inner regular on C.
We need to show that C satisfies the condition. Let A e C,
n
then A = 1A., A e fi disjoint. For each i, denote
i1 1 1
Ai â€œ Then Bj = ((Sj1 .tj1), (s'+l ,t'+1)]
belongs to 5: clearly A(C Int B for each i; hence
n n n
A  A C Int B â€” Int B. ) = Int B, denoting
i1 1 i1 1 i1 1
BUB.eC. Take A'  B.
We have now an extension o of o to C satisfying:
1) o is additive on C
2) o is regular on C
3) o has finite variation on C.
Then, by a standard theorem of measure theory, o can be extended
uniquely to a Borel measure m of finite variation. This measure
clearly coincides with a on 5, so the theorem is proved. B
Remarks.
1) The theorem proved by Radu is for Rn: we have restricted
p n
ourselves to r to enhance the clarity of the proof, but Rn presents
no additional difficulties (except with notation!).
77
2) The theorem holds in particular for the situation we use:
2
that where f is defined on R+, with f bounded, and f extended by
zero outside the first quadrant.
2
Suppose, now, that f is defined on R+, right continuous, with f
finite, and extend f by zero outside the first quadrant. As an
?
exercise, we shall compute explicitly the measure of some sets in R
using the limits developed in Theorem 2.3.3. More precisely, we will
compute the measure of points, intervals, and some rectangles in terms
of the "quandrantal" limits of f.
2
i) Let (s,t) e R . Denoting by the measure associated
with f, we compute mf({(s,t)}). We can write {(s,t)}  O An>
â– where A^ denotes the rectangle ( (p , q ), (p',q^) ] with
(Pn.Qn) < ^s,t) < (pn,qn^â€™ Pn+ + Sâ€™ qn + + tâ€™ pn+3â€™ qn+t (see F18ure
33). If we decompose A^ into four parts, labeled IIV in Figure 33,
q^) (s, t) , parts
A
n
! âœ“
1
I 11
t
1 I
1
1
1 III
(s , t)
IV
! /
(Pnâ€™qn}
Figure 33 Approximation of (s,t)(
73
therefore, consider the upper corner of A^ to be (s,t) for each n, so
that we can take = ((Pn>qfi),(s,t)] without loss of generality. We
have, then, by oadditivity of m^.,
m (((s,t)}) = lim m (A ) = lim A, (f)
f f n A
n n n
 lim (f(s,t)  f C p ,t)  f(s,q ) + f(p ,q ))
n n n n
n
= lim f(s,t)  lim f(p ,t)  lim f(s,q ) + lim f(p ,q )
n n n n
n n n n
since the individual limits exist by Theorem 2.3.3
 f(s,t)  f(s_,t+)  f(s+,t_) + f(s_,t_).
If we note that by right continuity we have f(s,t) = f(s+,t+), we see
that the measure of a point is analogous to the measure of a "halfÂ¬
open" rectangle, except we use the four 1lmlts to compute the measure
of a point.
ii) We next compute the measure m^, of intervals of the forms:
s) x (t,t'], {s} x [t,t']> (s) x [t,t'), {s} x (t,t'), and the
analogous "horizontal" intervals.
We begin with the closed interval I = {s} x [t,t']. We have
ob
I  C*} R , where R are rectangles of the form ((s ),(s'.t')]
^ , n n n n n n
n= 1
with (s ,t ) + + (s',t') + (s,t'). As before, we can take
n n n n
(s',t') = (s,t') for all n, so that R * ((s ,t ),(s,t#)] with
n n n n n
s < s, t < t, (s ,t ) (s,t) (see Figure 3**0 â€¢
n n n n
79
(s,t')
(s,t)
("s.tj
Figure 34 Approximation of an interval by rectangles
We have, then,
m (I) * lim m (R )
i f n
n
 lim(f(3,t')  f(s ,t')  f (s, t ) + f(s ,t ))
n n n n n
 f(s,t')  lim f(3 ,t')  lim f(s,t ) + lim f(s ,t )
_ n n n n
n n n
= f(s,t')  f(s_,t;)  f(s+,t_) + f(s_,t_).
Similarly, we can represent the interval J = [s,s'] x ft}
00
as J = R , where R  ((s ,t ),(s',t)], a < s, t < t,
^ ^ ji n n n n
(s^,^) t+ (St1) (Figure 35).
80
(snt)
(s , t )
n â€™ n
(s,t)
(s'.t)
(s'>tn)
Figure 35 Approximation of a horizontal interval
Again, we have
m (J)  lim m (R )
i in
n
 lim (f(s',t)  f(s',t )  f(s ,t) + f(s ,t ))
n n n n n
 f(s',t)  f(s',tj  f(s_,t + ) +
With these in hand, we can compute the measure of the halfopen and
open intervals: We write {s} x (t,t']  I \ (s,t)}, so
mf(fs}x(t,t ']) = mf(I)  mf(l(s,t)})
 f(s,t')  f(s_,tp  f(s + ,t_) + f(s_,t_)  (f(s,t)
81
 f(s+,t_)  + f(S_,t_))
 f(s.t')  f(s_,t')  f(s,t) + f(s_,t+).
Similarly, {s} x [t,t') = I\ {(s,C), so that
mf(ls} x [t,t')) = mf(I)  mf(((s,t')})
= f(s.t')  f(s_,t')  f(s+,t_) * f(s_,t_)
 (f(s.t')  f(s+,t_[)  f(s_,t') + f (s_, t'))
= f(s+it') f(s_,0 f(St,t_) + f(s_,t_).
As for the open interval, we have {s} x (t,t') =
Is} x [ t, t') \ {Cs, t)}, so
mf((s}x(t,t')) = mf((sjx[t,t'))  mf((s,t)})
= f(s + ,t')  f(s_,t_')  f(s+,t_) + f(s_,t_)  (f(s,t)
 f(s+,t_)  f(s_,t+) + f(s_,t_))
= f(s+,t')  f(s_,t')  f(s,t) + f(s_,t+).
We use the same method for the intervals with t fixed. We give the
results:
mf( [s,s')x(t)) = f(s\t+)  f(s',t_)  f(s_,t + ) + f(s_,t_)
82
mf((s,s']xt}) = f(s',t)  f(s+',t_)  f(s,t) + f(s + ,c_)
mf((s,s')x(t})  f(s',t+)  f(s_',t_)  f(s,t) + f(sf,t_).
p
lii) We now compute the measure of certain rectangles in R".
If we allow the possibility of each side being open or closed, this
gives 16 different rectangles, and we do not give explicit
computations for them all. We shall go into detail for only a few,
and indicate the procedure for the remainder.
First, we shall give the measure of an open rectangle
R =â– (s,s')x(t,t'). We write R Â«( ) R , where R  ((s ,t ),(s',t')]
n n n n n n
with (sn,tn) + (s, t), (s ,t ) t* (s',t') (see Figure 36).
r
L
(s,t)
Figure 36 Approximation of open rectangle
We have, then,
m (R) = Urn m.(R )
f f n
= lim (f(s'.t')  f(s',t )  f(s ,t') + f(s ,t ))
_ n n nâ€™ n n n n n
83
* lira f(s'.t')  lim f(s',t )  lim f(s ,t') + lira f(s ,t )
â€ž n n n n nn nn
n n n n
lim f(s'.t')  lim f(s',t )  lira f(s ,t') + lira f(s ,t )
n n n n , n n n n
s + ^s s + ts s +s 3 *s
n n n n
t'ttt'
n
t +t
n
t ttt
n
t +t
n
= f(s'.c')  f(s',t+)  f(s+,t') + f(s,t).
We next compute the measure m^. of a closed rectangle
R = [ (s, t), (s', t') ] = [s,s'] x [t,t']. We can write R , where
n
Rn  ^VV^n'^3 with (sn,tn) â™¦ â™¦ (s.t). * (s'.t'J
(see Figure 3*7). As before, when n Â» ", the rectangles labeled Iâ€”111
vanish, and so by aaddltivity of m. we can take (s',t') = (s',t')
i n n
without loss of generality.
Figure 3~7 Approximation of closed rectangle
84
He have
m (R) = lira m (R )
i n f n
â– lim(f(s',t')  f (s ,t')  f (3 ', t ) + f (s ,t )
n n n n
= f(s'.t')  lim f (s ,t')  lira f (s', t ) + lira f(s ,t )
S n t ttt n 3 t + 3 n n
n n n
t ttt
n
= f(s',t')  f(s_,t;)  f(s;,t_) * f(s_,t_).
With these in hand, to obtain the measure of other rectangles it is
simply a matter of adding or subtracting the appropriate intervals
that comprise the sides of the rectangle. We illustrate this
procedure (as well as check our wo"k!) by using R and some intervals
to compute m (((s,t),(s',t') ]). (Note that the rectangle
((s,t),(s ' ,t')]  (s,s'] x (t,t'].) We have, denoting this rectangle
by A (Figure 3'S),
Figure 38 The rectangle (s,s']x(t,t']
35
A = l([s,s']x[t,t']) \ ({s]x[t,t'])} \  (s,s']xt) } .
(Note that the points (s, t') , (s t) do not belong to A!) Then
mf(A) = mf([s,s']x[t,t'])  mf({s(x[t,t'])  m {(s,s ']xt))
= f(s'.t')  f(s_,t+')  f(s;,t_)  f(s_,t_)  (f(s.t')
 r(s_,t;)  f(st,t_) + f(s_,t_))  (f(s',t)  f(s;,t_)
 f(3,t) + f(s+,tj)
= f(s'.t')  f(s,t')  f(s',t) + f(s,t),
which is how m (A) was originally defined. The measure of other
rectangles can be computed similarly using the parts already
explicitly given.
3.2 Functions Associated With Measures
In this section we consider the converse problem, namely, given
p
an Evalued measure m on R with finite variation, is it possible to
associate a function with finite variation such that m  m in the
f
sense of Theorem 3.1.1? The following theorem provides a partial
answer to this question.
o
Theorem 3Â«2Â»1Â» Let m: B(R ) + E be a measure with finite variation
m. There exists a function f: R2 â–º E with Var_(f) < Â» on bounded
H
86
rectangles R such that ra is the measure associated with f by Theorem
3.1.1, l.e., such that for all rectangles R = ((s,t),(s',t')] we have
m(R) = AR(f) = f(s',t')  f(s',t)  f(s,t') + f(s,t).
Proof. Define f: R2 Â» E by f(s,t) = m((Â»,(s,t)] )* for
2
(s,t) e R . We show first of all that Â¿(f) = m(R) for bounded
R
rectangles R. We have, denoting R = ((s,t),(s',t')]:
AR(f)  f(s',f)  f(s',t)  (f(s.t')  f Cs,t))
 m(( â€”,(s',t')])  m(( â€”,(s',t)])  [m(( â€”, (s.t')])
 m( (<Â», (s, t) ]) ]
= (Â»,(s',t') ] N (Â»,(s',t)])  m((Â»,(s,t')] \ (=Â°, (s, t) ] )
 m( j (Â», (s ',t') ] \ (Â»,(s',t)]} \ l (Â°Â°, (s,t')] \ (Â»,(s,t)]})
= m(R).
We now show that f has finite variation on bounded rectangles,
i.e., that VarR(f) < Â« for bounded rectangles R = [ (s, t), (s', t') ].
Assume note: suppose there exists R  [(s,t),(s',t')] such that
VarR(f) = + â€œ. Denote by R the halfopen rectangle ((s, t), (s', t') ].
Let o: s = sQ < s1 < ... < = s' be a partition of [s,s'],
t: t = t < t < ... < t  t' be a partition of [t,t'], and let
0 1 n
P_ = oxt be the corresponding partition of R (cf. Prop. 2.2.5).
* Note: ( â€”, cS, t) ] = {z e R2;
z Â£ (s,t)).
37
Now, for any a>0, there exists such a partition P such that
VarR(fÂ¡P) > a, i.e., Z [A
0Â£iÂ£m1
0Â£jÂ£n1
However, the rectangles ((Sj,t^),(si +1 ,t )], 0 Â£ i Â£ m1,
0 Â£ j Â£ n1 are disjoint, and their union is contained in R,
so we have
0Â£iÂ£m1
0Â£jÂ£n1
 m(R).
Thus we have m (R) > q. a arbitrary > m(R) = + =Â», a contradiction
since m has finite variation. Hence Var (f) < Â» for R bounded, and
R
the theorem is proved.
Remarks.
1) We have said nothing about uniqueness of f. In the case of
functions on the line, f is determined within a constant by m (i.e.,
any other associated function g is determined by adding or subtracting
a constant from f) , but here this is not the case. In fact, as we
have seen before (Example 2.1.2) that many completely unrelated
functions can have zero as its associated measure.
2) The oadditivity of m implies that f is incrementally right
continuous, but as we have seen in Chapter II this is insufficient to
imply right continuity in the order sense without imposing finite
variation on the onedimensional paths f(s;) and f(*,t).
98
p
We return now to the situation with f defined on R right
continuous, with finite variation f, both extended by zero outside
the first quadrant. We have an important equality we shall make use
of in the next chapter, which is given in the following.
Theorem 3.2.2. Let m be the Evalued measure associated with f, and
let m  j be the realvalued measure associated with f. Then m^. has
finite variation m^, and we have the equality
m
m "
Proof. We showed in Theorem 3.1.1 that m has finite variation. The
real thing to be proved here is the equality.
Let S be the semiring of rectangles of the form
R = ((s, t) , (s ', t') ] . We shall show first that m  ^ j =  m^,  on S.
We must consider various cases.
First of all, if (s',t') < (0,0), then = mf(R) = 0.
We will assume, then, in what follows, that (s',t') lies in the first
quadrant. There are four cases, according to what quadrant (s,t) lies
in.
1) Assume (s,t) lies in the first quadrant. Let o: s =
sQ < si < ... < s^ = s' be a partition of [s,s'], t: t  t < t
< ... < t  t' be a partition of [t, t ' ]. Denote P = nxi the
n 
corresponding partition of R:
 = lRi,jlRi,J â€œ [(3iâ€™V'(si*rVl)]â€™ 0 < i < m1, 0 < j < n1 l
Denote by R. . the corresponding halfopen rectangles
89
((Sj,tj),(s^ + i ,t )]. (We shall use this notation in the other oases
as well.) We have
E m (R )  Z A (f)  Â£ Var (f)  A (f[)
i.J â€™J l.J "l.J R
(cf. Remark 9 following Defn. 2.3.1). and the right hand side is just
mij,(R). The family (R^ ) forms a disjoint cover of R with
R, . â– = R, so taking supremum we obtain m (R) S m. . (R). For
q 1 ,j i ir.i
other inequality, let e>0. There exists a grill oxt such that
the
Z A (f)  > Var (f)  a  m, . (R)  e.
l.J Ri.j R f
But the left hand side is equal to Z m (R. .), and the R
i.j 1,J 1,J
forms a decomposition of R, so we have
m (R) 2 Z m (R ) = I IA (f) > m. (R)  e,
f l.J f iJ i.j Ri,j f
i.e., mf(R) > m  j.  (R)  a. Letting e Â» o (neither side now depends
on the corresponding grill), we obtain mf(R) i mf(R); hence
mf(R)  m  f j(R).
2) Suppose now (s,t) lies in the second quadrant, i.e., s<0,
t20. For any grill oxt, we can refine a by including zero if it is
not already there, so that we may take oxt with zero included in a,
and compute variations with these grills (Figure 39). Denote by k
the index where s = 0: For i < k  2, m.(R ,) = 0; we also have
k " f 1, j
90
Figure 3"9 The grid axx
Wi.jâ€™1 * *f(sk'tJ+i5 â€¢ fÃvV â€¢ 'â€˜ViVi1 + f(Viâ€™Vi
= f(0,tJ+1)  f(O.tj).
Putting everything together, we get
_ n1
E !>l = I f(0,t )  f(0,t ) * I A (f)(
i,j ,J J1 J J i2k "i.j
jSO
var[t(t.]f(Â°..) * Var[(0it)i(3.it.)](f)
 m i f(R),
since ra  j. j (R)  f(s',t')  f(s',t)  f(s,t') + f(s,t)
  f  (s' ,t')  f(s',t) = [f(0,0) + Var[0 ^fi,0) + Var[0 t.]f(0,O
+ Var[(0.0),(s',t')](f)]  [lf(Â°â€™0)l + Var[0,s *]f (* â€™0) + Var[0,t]f<0â€™Â°
+ Var[(0>0)1(s',t)](f):l = CVar[ (0,0) ,(s',t')](f) ' Var[(0,0),(s',t))(f)]
91
[ Var
[o,t']f(0,') * Varâ€™[o1t]f(0,â€˜)] â€œ Var[(o,t),
â™¦ Var. , f(0,O. Taking supremum over grills oxx on both sides, we
L vÂ»t J
get (as before) ]mf(R) Â¿ mf(R).
On the other hand, for any e>0, there exists a partition x ' of
n1
C t, t' ] such that z f(0,t )  f(0,t ) > Varr ,,f(0,O  Â§ , and
j_q J + ' J L t  t/ J Â¿
a grill OqXTq of [0,s']x[t,t'] such that IjAR (f) >
i. j
Varr,_ . , , , ,,,,(f)  % . We choose a common refinement t of t '
L(0,t),(s ,t )J 2
and tq, and extend arbitrarily to get a partition o of [s,s'].
Then, for the grill cjxt, we have
n1
Â£ m (R )  l f(0,t )  f(0,t )  * l IA (f) (as before)
i.j j0 J 3 a xt "i,j
> Var[t,t']f(Â°'
2 + Var[(0,t),(s',t')](f)
Â£
2
* ltf  (R)  E.
Since the left hand side is bounded above by m (R) for any grill,
we have mf(R) > mj f  (R)  e. Letting e 0 again, we get
mf(R) S nij(R). hence equality.
The next case proceeds similarly.
3) (s,t) in the fourth quadrant: siO, t<0. This time, we
refine t by including zero if necessary, and compute variations along
such grill3. Denoting t  0 a; before, we have (same computation as
before) :
92
m1
2 m (R )  2 Sf(s 05  f(a .0)  + I \t (f)
i,j 1=0 i>0 H1,J
jÂ£k
< Â¥arr ,,f(0,)
[s.s J
Var . , .(f)
L(s,0),(s ,t ) J
mlfl(R)
(same as before). Taking supremum we get m,,(R) Â£ m  ^.  (R). The
proof of the other inequality is the same as that for case (2).
H) Finally, assume (s,t) < (0,0). We proceed similar to the
above, but this time we add zero to both o and t, and use these
partitions in our figuring of variations (Figure 3_10). Denote
s. = 0, t,  0. For i < k  2 or j Â£12, we have lm.(R, .)! = 0.
k 1 " f i 1
Figure 3~10 The grid oxt
For i â– k  1, j â– 1  1, we have [[m^^)Â¡ = f(0,0), for
93
i = k  1 , j i 1, we have m,(R. )l  f(0,t, ,)  f(0,t.). For
I i,j J+l j
j = 1  1, i Ã k we have mf(R1 )  f(si+1,0)  f(s.,0), all as
before. Putting everything together, we have
m1
r m (r ) = [f(o.o)  + r If(a ,0)  f(s.,o)
l.j i=k 1
n1
+ E f(0.t )  f(0,t ) * E Aâ€ž (f)
j1 J 1 J iSk Rl,j
m
< f(0,0) â™¦ Var[0>g.]r(.10) â™¦ Var^ .]f(0, â€¢)
Var[(0,0),(S',t')](f)
f(3 ',t') = mf(R).
Taking supremum again, we obtain m^,(R) Â£ mi^,(R). The proof the
other direction is similar to the ones before: for e>0, we choose a
common o,t so that
Zf(s1+1,0)  f(s1,0) > Var[Q s,]f(.,0)  
T]f(0,t.+1)  f(O.tj) > Var[0>t.]f(0I.)  f , and
E jiR (f)  > Van
OXT l,j
[(0,0),(s',t')](f) ' 3 â€¢
We extend o and t arbitrarily to partitions o',t' of [s,s']
9Â«
and respectively. The same computation as before gives
I gm (R ) = f(0,0) + If(s ,0)  f(s 0)1 + if(0,t >
i.j o t J
 f(0,t ) + l AR CfJI
OXT l,j
> f(0,0)+Var[0i3.]f(..0) + Var[0>t.]f
+ Var[(o,o),(s',t')](f) ' 3
 f(s',t')  e  m.fi(R)  e.
Hence nif CR) 
K(R> I a mf
> mf(R)  e. Letting e
(R); hence equality.
0, we obtain
This takes care of all the possibilities, so we have mf = m^,
of S. Moreover, both are oadditive on S; the first since is by
Theorem 3.1.1, the second since f is right continuous by Theorem
2.3.2. Since , m^j are equal and oadditive on S, they are equal
on o(S) Â» B(R^), and the theorem is proved. B
CHAPTER IV
VECTORVALUED PROCESSES
WITH FINITE VARIATION
An important part of the general theory of processes in one
parameter is the correspondence between processes of finite variation
and measures on R+xP (see for example Dellacherie and Meyer [5, VI.
6989] and also Kussmaul [10]). This correspondence finds
applications in the notion of dual projections of processes, which are
used in the theory of potentials and in decomposition of
supermartingales (see for example Dellacherie and Meyer [5, nos. VI.
71113], also Rao [17] and Metivier [11]).
In the oneparameter case, the extension of the correspondence to
Banachvalued processes Is done in Dellacherie and Meyer [5]. In two
parameters, the correspondence for realvalued processes is stated
(more or less) in Meyer [12]; we shall presently give a more directly
applicable (for our purposes) version, along with a proof, as the case
2
of finite variation on R+ is more delicate (as we have seen). In
fact, many times, in the literature results are given for increasing
processes, and then extended by defining a process of finite variation
as a difference of two increasing processes. The method we use here
is a little more constructive.
95
96
4.1 Definitions and Preliminaries
Throughout this chapter, (fl,F,P) will denote a complete
probability space, (F ) a filtration of subofields of F
Z zeR2
+
satisfying the usual conditions. We also assume (F ) satisfies the
axiom (FA) of Cairoli and Walsh [2] (see section 1.2). Throughout
2
this chapter we shall denote by M the product ofield B(R+)xF. We now
state some definitions we will use in this chapter. (Some are
restatements from Chapter I, but we will give them again here for
completeness.)
Definition 1.1 .
a) A (twoparameter) stochastic function is a function (not
o
necessarily Mmeasurable) X defined on R^xfl. Here, X will have values
in a Banach space, usually either in a Bspace E, or in the space
L(E,F) of continuous linear maps from E into another Banach space F.
We will consider X extended by zero outside the first quadrant, as we
2
did for functions defined on R .
+
b) A (twoparameter) stochastic process is a function
2 2
X: R+xÂ£i Â» E, measurable with respect to M = B(Râ€˜)xF. A process X is
2
called adapted if X : a â€¢* E is F measurable for each z c R (see
â€” w Z Z +
Millet and Sucheston [13] and Chevalier [3] for related notions). We
generally use the term raw or brut to refer to a process that is not
necessarily adapted, l.e., such that X^ is Fmeasurable for each
2
For fixed w e Â¡2, the map X (w): R+ â– + E is called a path of the
process. Each path is a function defined on the first quadrant, so we
97
snail use the results from the earlier chapte"S in studying these
processes. In particular, the variation of a process is defined in
terms of its paths. We have the following definitions.
Definition 4.1.2.
a) Let X be a raw process. We call X a process of finite
p
variation if, for each w, the path X_(w): â– * B is a function of
finite variation in the sense of Definition 2.3.1. We define, for X a
process of finite variation, a realvalued process JX, called the
variation of X by the following:
2
for w e n, z = (s,t) e R+,
X (w) = X_(w)(s,t)
â– lXC0,0)(w)l + Var[0,s]lx.(w)Icâ€˜1Â°) + Var[Ojt]X.(w)(0,.)
+ Var[(0,0),(s,t)](lx.(w)l)
b) If the random variable Â¡X^  lim x. Â£ + â– (which
3Â»â€ â€™
t>Â»
exists since x is increasing in the order sense) is Pintegrable, we
say X has integrable variation.
In this chapter, we shall concern ourselves with processes of
integrable variation. We will consider them extended by zero outside
?
R , as we did for functions earlier.
Remark. In the book by Dellacherie and Meyer [5], processes of finite
variation are defined as differences of increasing processes. In two
parameters, it seems we might have a problem with this, as we have two
98
distinct definitions of "increasing." However, we have shown (Prop.
2.3.'O that a process of finite variation, as we have defined it here,
can be written as a difference of two processes (apply Prop. 2.3.H to
each path) that are increasing in both senses, thus removing the
ambiguity.
We give now one more result concerning functions, which will be
used extensively in later theorems.
Proposition J4.1 .3. If g: R^ > L(E,F) is a function with finite
variation g (Defn. 2.3.1), then for every x e E and z e F', the
2 2
functions gx: R+ *â– F and : R+ Â» R have finite variations gx
and  . (For the realvalued functions we shall use double bars
O
for the absolute value to avoid confusion.) Moreover, if f: r Â» r
+
is dgintegrable (i.e., d  gintegrable) on a set ICR^, then f is
d(gx) and d integrable on I, and we have
(â– 1.3.1)
x/jfdg = Jjfxdg = jjfdCgx)
and also
(â– 1.3.2)
= = JIfd.
2
Proof. For the first assertion, let z = (s,t) e R . We have, from
Definition 2.3.1,
+ Var
[(0,0) , (s,t)]
(g) < â€œ.
We have (gx)(OtO)J * g(0,0)xj Â£ g(0,0)x, and from Che oneÂ¬
dimensional case proved in Dinculeanu [7], we have
Var,[0,s](gX)(''0) â€˜ ' Var'r0,s]g(',0):
Var[o,t](8x)(0â€™) 5 ^ ' Var[0,t]g(Q,')*
In fact (we prove the first; the proof of the second is identical)
for 0 = sâ€ž < s, < ... < s = s a partition of [0,s], we have
0 i n
(gx)(s ,0)  (gx)(3 ,0) = (g(s.+1,0)  g(s.,0))x
Ã Jg(sl + 1,0)  g(si>0)  x
for i=0,1,2 n1 . Summing over i, we obtain
n1 n1
1 IKsxMs ,0)  (gx)(s ,0)  Â£ l xâ€¢ g(s ,0)  g(s. ,0) 
i0 i=0 1
n1
38 x[ I g(s.+1 Â»Â°) â– g(s.,0)
i = 0
 Â»XVar[0,s]g(â€™'0)
Taking supremum over partitions of [0,s], we get Var. ,(gx)(,0)
L U , S J
Â£ x*Var0 sjg( â€¢ ,0). The same proof gives Var^ t j (gx) (0,Â»)
Â£ xVarj.Q ). We obtain a similar Inequality for the
remaining term of  gx : For any rectangle R = [(p,q),(p',q')]C! R
we have
H&R(gx) = (gx)(p',q ')  (gx)(p ',q)  (gx)(p,q ') + (gx)(p,q)
100
*â– [g(p'.q')  g(p'.q)  g(p.q') + g(p.q)]x
" (iRg)xH
Â£ MiRg
Then, for any grill oxt on [(0,0),(s,t)], we have
Z SACCs t ) (s t )](gx) I S Ix! E IAr (e5!
3XT ,lS1+1,tJ+1)j aXT Klfj
: Var
[(0,0),(s,t)]
(g)
Taking supremum over grills of [(0,0) , (s,t)] , we obtain
Var[(0,0),
together, we get
j gx  (s, t) = ( gx) (0,0)  + Var>Q gj (gx) (â€¢,0) + Var0 t^ (gx) (0 , â– )
+ Var[(0,0),(s,t)](gx)
Â£ x11g(0,0)  + xVar[0)S]g(,0) + x*Var^t]g(0,â€¢>
+ "xVar[(0,0),(s,t)](g)
* xg(s,t),
so  gx  (s, t) S ] x 11 g ] (s,t) < â€œ for all (s,t) e R^, i.e., gx has finite
variation. The same argument works for : in fact, for
(p,q) e R^, we have (p,q) = <  (gx)(p,q)  z [
i g(p,q)xz. The same proof as above then gives
101
I  (s,t) S x[zg(s,t) < Â»
2
far all (s,t) e R+, l.e., has finite variation for all
x e E, z e F'.
Now to prove equalities (4.3.1) and (4.3.2): Let I e B(R^) ,
2 i r
f: R + *â– R be dgintegrable. Assume g(I) < Â®. We shall use the
monotone class theorem 1.5.2 to prove the equalities for bounded f,
then extended to fintegrable.
Let H denote the set of bounded, realvalued, dgintegrable (on
I) functions f satisfying (4.3.1). Then:
i)H is a vector space (evidently),
ii)H contains the constants: let f = a constant. Then
x/jfdg  xjjadg = x/aljdg = x(ag(I)) = (ax)g(I)
(g(I) is the measure of I for the measure dg),
Jjfxdg  JtcbOljdg = (ax) gti) , and
Jjfd(gx) = /aljdfgx) = a*(gx)(I) = a(g(I)x) = (ax)g(I).
Hence xj^fdg = /jfxdg = /jfdigx), which is (4.3.1).
iii)H is closed under uniform convergence: suppose f Â»â€¢ f
uniformly and (4.3.1) holds for each f . Then
n
x/jfndg Â» x/^fdg, since by Lebesgue dominated convergence we
have that f is dgintegrable over I and Lf df + Lfdg,
â€¢'In â€˜ I
hence x[,f dg â– + xLfdg. (In fact, from some index n on we
11 n Â° 1 I B 0
â€™02
have If f < i => for n Â£ n. we have If I < If I + 1 ,
â€™ n ' 2 0 * n' 1 Hq1
which is dgintegrable on I.) Similarly, f x â– * f*
n
uniformly, so from some index nâ€ž on, If xfx < 1 =>
0 n 1 Â¿
for n>nQ, fnx < f^x * 1 S f x + 1, integrable
since H is a vector space. By Lebesque, then,
Jjfnxdg * Jjfxdg. Finally, since gx < gx, fR
are d(gx)integrable, so J^f d(gx) â– * JjfdCgx) by Lebesgue as
before. For each n we have xLf dg = Txf dg â€¢ j,f d(gx),
JI n ; I n JInÂ°
so on passing to the limit we get xj fdg > JjXfdg = JjfdCgx),
which is (14.3.1).
iv) Let (f ) be a uniformly bounded increasing sequence of
n
positive functions from H, and denote f = lim f . Show
n
n
f e H. Let M Â£ f for all n. Then we have
n
Jlfndg Ã¡ JjMdg * Mâ€¢gCl) for all n. By Lebesgue, f is
dgintegrable on I, and J^f^dg /fdg, hence
x/ifndg  x/jfdg. Similarly, fnx Ã fnx
S Mx e L1(dg), so fx is dgintegrable on I by Lebesgue,
and l^f^xdg * fxdg. Also, f is d(gx)integrable since
M and JIMd(gx) < M[xÂ¡g(I), so by Lebesgue again
103
we have f is d(gx)integrable on I and Lf d(ex) *
'I n
JjfdCgx). Now, (4.3.1) holds for each n, so passing to
limits as before, (4.3.1) holds for f as well.
Now, let S be the family of sets of the form R = ((s, t), (s', t') ]
r) R^, Since the rectangles ((s,t),(s ',t')] form a semiring
2 2
generating the Borel ofield on R , S is a semiring generating B(R+).
Let, then, C be the family of indicators of sets of S. To complete
the monotone class argument, we must show that CCH and that C is
closed under multiplication, as H then contains all bounded functions
measurable with respect to o(C)  B(R^).
C is closed under multiplication, as 1D 1  1D _ , and S is
R1 R2 R1OR2
a semiring, so R D R e S => 1 e C. Now we show that (4.3.1)
1 Â¿ tyt k2
holds for f = 1 , R e S. We have
K
x^i1Rdg â€œ xC1Rmdg = x(g(RHi))
(Again g() refers to the measure dg.) Since x(g(R Pi I))  <
x  â€¢  g  (R O I) Â£ fxMgl(I) < â€, 1 is dgintegrable and d(gx)
integrable. Also, JjXlRdg = /x1Rn].dg = xlglRHl)), and
I1Rd(gx) = /lRnid(gx) = (gxMRPiI)  xlglROl)), hence
x/j1 rt*g  JjXlpdg = jj^digx), which is (4.3.1), so C H. This
completes the proof for f bounded, g(I) < Â».
Assume now, g(I) < Â», f dgintegrable on I (not necessarily
bounded). There exists a sequence (f ) of bounded functions
104
converging to f a.s. and in L1(d[g) on I, with (4.3.1) satisfied for
each n. In the first integral, we have J^f^dg * J^fdg, hence
+ xJIfd8. Also, /Ixfndg  JjXfdgl = /Ix(fnf)dg
S IxliIlfn'flds + 0; hence xj^dg Â» xfjfdg. Finally, f is d(gx)
integrable [6, Theorem 4, p. 172], and we have [Jjf^digx)  /Tfd(gx) 
* JI(fnf)d(gx) < /j.rnrd( g x) . 1 xJxfn*f d(g) * 0 as
n * â€œ. Then Jjf^dCgx) *â– J fd(gx). Since (4.3.1) holds for each n, we
have it for f as well by passing to the limit.
2
Finally, let I e B(R+) , f dgint enable on I. There exists a
p
sequence (I ) of sets from B(R ) with g(I ) < â€, and I + I.
n + 1 1 n n
Then f* 1 j. *â€¢ f1j a.s. and in L1(dg) and L1 (d( g( xJ)) by Lebesgue
n
(since [flj  Â£ f â€¢ 1 j j e L1 C d  g I ) and L1 (d]g xJ)); (4.3.1) is
n
satisfied for each 1^, so we pass to limits exactly as above. This
completes the proof of (4.3.1). The proof of (4.3.2) is completely
analogous (since  Â£ Ix!lzg). I
We conclude this section with the theorem establishing the
correspondence between stochastic measures (Pmeasures) with finite
variation and processes of integrable variation for the case of realÂ¬
valued processes and measures. Although this is a special case of the
more general result we will establish later, it cannot be deduced from
that, since our proof for the vectorvalued case will make use of the
realvalued result.
Theorem 4.1.4. There is a onetoone correspondence X ** between
raw processes X: R+xÃi * R with raw integrable variation jX and
105
stochastic measures y^ with finite variation [y , given by the
equality
(4.4.1) yâ€ž(A) = E(Ã A dX ) for A bounded, measurable.
X 2 Z Z
K
4
Remark. We shall later prove the equality y, . = y  for X with
values in a Banach space, from which the equality follows for realÂ¬
valued X as a special case.
Proof. We remark first that the correspondence is onetoone in the
sense that we identify processes that differ only on an evanescent
set.
2
1) Let X: R+xÃÂ¡ â– * R be a raw process with raw integrable
variation x. For any bounded measurable procesa A, a[ Â£ M, the
map w + Ã A (w)dX (w) is in L1(P). In fact, we have
V z 2
+
f _A (w)dX (w) < { A (w)dx (w) Â£ MIXI (w), and by assumption
RÂ¿ z z RÂ¿ z z
lX_ c L1(P). Then, for any M e M = B(R2)xF, E( 1MdX ) exists.
Set
, then, yâ€ž(M) = E(i 1 dX ). Then, for step functions
x 'M z
n n
B  I a 1 â– E a, e R, we have yY(B) = jBdy  E a y(M.) =
, j 1 il , 1 1 X ..11
1=1 l 1=1
,Vi(E(iR2(v w5 â– / Â«i/iVâ€™/v â– E<
1=1 1 1=1 R 1 1=1 R 1
E(f ( E a,1â€ž ) dX ) = E(f _B dX ). Now, suppose A is bounded,
R2 i=l 1 "i Z Z R2 2 2
1 36
measurable. Let An be a sequence of step functions, converging
uniformly to A except on an evanescent set. For each n, we have
yv(An) = E( Ã â€žAr'dX ). Since A is bounded, E( fA dX ) exists, as we
X Â¿ z z J z z
R+
have shown. Moreover,
E(f A^Xz)  E(J AzdXz)  E(i (AnA)zdXz)
R R R
< E(J 2AnAzdxz)
< (sup AzAz) X j m  0
ztR ^
as n â€¢ by uniform convergence. Thus E( AndX ) E( ÃA dX ). For
J z z 1 z z
each n, the double Integral is equal to vx(An) , and p^(An) *â– ux(A)
(since has finite variation, which we shall prove in a minute),
hence we have equality in passing to the limit, i.e.,
PXCA)  E(/R2AzdXz).
Now, we must show that (1) p^ is a stochastic measure, and (li)
p^ has finite variation. The first is easy: if M e M is evanescent,
then M(w) = 0 for almost all w, hence
Ã
R
dX (w)
2 M z z
0 Pa.s., so E(f(i ) dX ) = o,
1 M Z Z
107
i.e., ux(M) = 0. As for (ii), let M e M, (M^) 1  1,2 n disjoint
n
sets form M, with M. We have then Z p (M )  
1 1 i1 X 1
51 iE(/ 2(1â€ž >zdV I * E E0 2(1M )zdlXlz) " E( E I ?(1M )zdlXlz) *
i1 Z i1 Z 2 i1 R^ Mi 2 2
E(f o'1!) u ) dlx!) since all sections are disjoint = E(f _(1â€ž) dlxl ).
â€˜2 M. z 1 ' * â€˜2 M z 1 'z
The last integral is independent of the family (M.), so by taking
supremum we get n (M) Â£ E( f (1 ) dIXI ) Â£ E(IX! ) < ", so yv has
X1 Jâ€ž2 M z z 1 1â€ X
finite variation (in particular yâ€žl is bounded by E(IXI )).
Uniqueness of u is evident: this completes one half of the
correspondence. Next we prove the converse.
2) Let y be a stochastic measure with finite variation y.
Assume, first, yiO (then y  y). We will associate an increasing
process X (increasing in both senses) satisfying (A.A.I); then the
final result is an easy consequence of the decomposition of measures
with finite variation.
2
For each bounded r.v. Y and u e R+, consider the raw process
YU(w)  Y( w) â€¢ I. ,(z). The map A defined by A (Y) = y(YU) =
z L u ^ u j u u
u(Y*I^ u) is a bounded, positive measure on (Q,F). In fact: for
B e F, A (B)  y (1 â– 1 r .) = y(Bx[0,u]) Â£ y(nx[0,u]). Also, A is P
U D L U y U J U
absolutely continuous; if P(B) = 0, then X^(B) â– u(Bx[0,u]) = 0 since
m is a stochastic measure. Then \ has a density a with respect to
u J u ^
p
108
Now, if uÂ£v, then A^ Â£ A^ (since for B e F, A (B) =
p(3x[0,u]) Â£ y (Bx[ 0, v]) = A (B)) ; hence a Â£ a a.s. We set, for w
v u v
2
running through the rationals in R + (i.e., points with rational
coordinates), a1 = sup a . a1 is increasing* in both senses of
z . w
wÂ£z
Definition 2.1.3, as we now show.
The order sense is no problem, as we are taking supremum over a
bigger set for bigger z. For the other, let (s,t) < (s',t'), denote
R  ((s,t) , (s',t')]. First of all, for any such rectangle R, we have
AR(a) 2 0 a.s. In fact, for B e F, E(1gAR(a))  E(1Da=.^,) 
B s t'
E(1 a )  E(1a ) â™¦ E(1 a . ) = A , <(B)  A .(B)  A ,(B) +
rist B Sl B St St St St
X (B) = u(Bx[(0,0),(s',t')])  y(Bx[(0,0),(s',t)]) 
y(Bx[(0,0),(s,t')]) + u(Bx[(0,0),(s,t)]) = p(Bx((s,t),(s ',t')]) > 0.
Since the four functions that make up A (a) are Fmeasurable, we have
n
AR(a) i 0 a.s. (I.e., for w t N) for R with rational coordinates.
Now, we show that for w t N (see note preceding page), we have
1 2
Ad(o (w)) 2 0 for all R = ((s,t) , (s',t')]C R . Let w t N and suppose
K +
* Outside an evanescent set: for uÂ£v, u,v rationals, a Â£ a a.s.,
so for each pair u,v there Is a negligible set outside of
which ci^iw) Â£ av(w). We put these together into a common
negligible set N outside of which a (w) Â£ a (w) for any u,v
2 u v
rationals in R .
+
109
Ap(ct (w) ) < Â£ < 0. We can find, since a (w) is increasing (on the
rationals) on this set, ratlonals p
Figure 91) so that we have ajg. fc,j(w)  a(p, ,^(w) <   ,
â€œIt'(w) â€˜ apq'(w) <  t â€™ etCâ€˜ (re0311 Â£<0) If A((S,t),(S',t')](a1Cw))
< e, then we have
A,, , , . , . (ct(w))  a , ,(w)  a , (w)  a ,(w) + a (w)
((p,q), (p ,q )] p q p q pq pq
ab,..(w)  a , (w)  a .(w) + a' (w)
st p q pq st
(by definition of a"')
Â£ Vt'Cw) " (Vt{w) + " (aIt(w)+ + v(w)
A((s,t),(s',t')]Ca (m)) " 2 < E " 2
 < 0,
i.e., A,, . , , ,,,(a(w)) < 0, a contradiction. Then a' is
((p,q),(p ,q )J
increasing in both senses for w t N.
2 1 1
Now, for each z e R , we have a.s. a < a < a . Also, the
+ z z z +
map z * \ (Y) is right continuous. In fact, lim X (Y) =
u+z u
lim u(Y(w) â€¢ I, ,(v)) = p(Y(w) â€¢ Ir ,(v)). Then
u*z LÂ°.u^ CO.z]
Xz = a^+ is also a density of \ with respect to P. In fact, on one
hand, since az < a^+, for B e F, *z(B) = E(1Baz) < E(1Baz+) =
no
E(1nX ). On the other hand, E(1 X ) â– = E(10a' ) = E(1.(lim a1))
u+z
= lim (E(1 a')) (by monotone convergence) Â£ lit (E(1 a )) =
D U B U
u+z utz
lim X (B) = \ (B) by right continuity. Putting the two together,
u+z
A (B) = E(1 X ) for B c F, i.e., X is a density for X .
z B z z z
Then for processes of the form A^(w)  Y(w) â€¢ 1^ u(z).
u(A) = E(f A dX ), since E(f A dX ) = E(f _Y(w) â– Irâ€ž ,(z)dX )
o2 z z D2 z z n2 [0,u] Z
E(Y(w)J I dX ) = E(Y(w) â€¢ X ) = X (Y)  y(Y â€¢ Irâ€ž .) = y(A).
J LD.uJ z u u [0,u]
Moreover, a * lim a , so a is also (same proof)
z+ z z+
u + z
u rational
incrementally increasing for w t N; hence X = a1 is an integrable
z z+
increasing process: more precisely, x^g ^(w)  XjQ Q^(w) +
X, + ,>(") + x, fdâ€œ)  *iX. (w) for w t N, so outside
(s,0) (0,t) (s,t) (s,t)
( 2
an evanescent set XJ < Ap(R+xQ) (we shall use this later). To prove
the equality (t.A.1) for A bounded, we use a monotone classes
argument.
Let H be the set of all bounded, Mmeasurable processes A (w) for
z
which (A.A.1) holds. H is clearly a vector space. Also, H contains
the constants: If A = c constant, then A = lim c â€¢ , , ,_,(z),
[(0,0),(n,n)]
and we have shown (4.4.1) for these processes already. We get the
result in the limit by monotone convergence. More precisely, denoting
An(w) = c â€¢ I, nN , ,,(z), we have> for w outside a negligible
set, supi An(w)dX (w) = supX (w) < Â», so by monotone convergence
we have \ An(w)dX (w) â– + j A (w)dX (w) Pa.s. Then, the maps
_ii z z s z z
R+ R+
Â» â€¢ â€žAn(w)dX (w) increase to w *â– f A (w)dX (w) Pa.s. and the
V z z R2 z z
+ +
latter is integrable as we saw above, so by monotone convergence we
have that E(/ AndX ) E(J A dX ). Since u(An) > W(A) and (4.4.1)
R2 Z Z R2 Z Z
+ +
holds for each n, (4.4.1) holds in the limit as well. Also, H is
closed under uniform convergence: let A be bounded, measurable, An a
sequence from H with An * A uniformly. By Lebesgue p(An) â– + p(A).
Also, for almost all w, X(s) is a bounded positive measure, so
I â€žAn(w)dX (w) â– + Ã A (w)dX (w) Pa.s. (by Lebesgue again) so by
R2 z Z R^ z
Lebesgue the map w > j ^A^twJdX^Cw) is Pintegrable (since each
An e H) and EC f 0AndX ) â– + E(i A dX ). Since (4.4.1) holds for each
'2z z J 2 z z
R+ R+
n, it holds in the limit as well. Finally, let An be uniformly
bounded, AntA, An e H for all n. As above, using a double application
of the monotone convergence theorem this time, we have E(Ã AndX )
jd2 z z
> E(f A dX ) , and u(An) â– + p(A), and we conclude as above,
z z
+
To complete the monotone class argument, let C be the class of
processes of the form Y(w) â€¢ I ,(z). We already know C iÃ¼ H; it is
L U, U J
easy to see (taking Y indicators of sets of F) that o(C)  M.
Finally, C is closed under multiplication; in fact,
(Y,(w) â€¢ I[0iU](z))(Y2(w) â€¢ I[0>v](z))  (Y1(w)Y2(w)) â€¢ I[0>uv](z)
e C, and the monotone class argument is done. This completes the
112
proof for y i 0; if y has finite variation, we write y = y  y ,
+ + â€” â€”
and associate X with y and X with y . We have, then, for A
bounded, measurable: u(A) = u+(A)  y (A) = E(Ã â€žA dX+) 
R2 z z
+
E(/_2VV  â– / ?W â– E(/ Ad(xÂ¡  V> Setting
R+ â€œ â€œ r; â€œ " r; â€œ â€œ r;
x = X+  x", y(A) = E( / 2AzdXz), and x = X+  X_ <  X Â¡ + x",
so X has integrable variation, and the theorem is proved. I
Remark. In Meyer [12] a version of this theorem (without proof) is
2
given for Pmeasures and random measures on R x!1.
4.2 Measures Associated With VectorValued
Stochastic Functions
In this section we shall show that, starting with a stochastic
function, we can associate a Pmeasure, with finite variation if the
function has integrable variation. Our first theorem is for
measurable processes with integrable variation.
Theorem 4.2.1. Let E be a Banach space, X an Evalued, raw, right
p
continuous process such that X^ is integrable for every z e R+,
with raw integrable variation
lxl(s,t) lxco,o)! + Var[o,s](x(,o)) + Var[o,t](x(o,))
+ Var[(0,0),(s,t)](X)
There is a stochastic measure (Pmeasure) y^: B(R+)xF â– + E with finite
variation satisfying the following.
113
If * is any scalarvalued measurable process, we have
4> e L1(ux) if and only if E(f ?i> dxz) < Â». In this case,
R
+
E(/ _4> dX ) is defined,
R2 Z Z
(4.2.1) uY(*) = E(f i dX ), and
R2 Z Z
+
(*4.2.2) ux(iO  Elf 2*zdXz),
i.e., ux  vx.
Proof. For Me M, the integral E( f 1 dX ) is defined; to see
â€” V M z
+
this, we use a monotone classes argument. More precisely, let
H = Â¡M e M: E(f â€ž1â€ždX ) is defined]. We will show that H is an
' 2 M z
n
+
algebra, is closed under monotone convergence, and contains a semiring
generating M; we then conclude from the monotone class theorem that
H 3 M, hence H = M.
H is an algebra. Let A,B z H; show AiJB, AÃ^IB, AC e H. First of
all, J 1 (w)dX (w) exists Pa.s. as well as Ã 1 (w)dX (w).
RÂ¿ A z RÂ¿ B z
Then 1A(Â«) â€¢ 1R(w) is dX^Cw)integrable almost surely, i.e.,
f ,,1A(w) 1 1g(w)dX (w) = { 21ApB(w)dX (w) exists Pa.s. Moreover,
R+ R +
1/ 21AAB(w)dXz(w)l * K1AOB(w)dlXlZ(w) * / 21A(w)dlXlZ(w): hence
R R + R+
w * f 21 Ar\B^dXz^ iS plnte8rat)le, i'e> Â£(J 21AOBdXz^ exlsts>
111)
so *Ob e H. We get A (J B e H by writing 1 ^ = 1ft + 1g  1^^g
and AÂ° by 1 >11, (E< f1dX ) = E(X )).
,c A J z <â– >
A
H is closed under monotone convergence. Let A^ be a sequence of sets
from H with A increasing to A. For almost all w,  .1 . (w)dX (w)
n '2 A z
R+ n
/ 21A(w)dx2(w) by Lebesgue (since 1, Â£ 1 e L1(x(w))). For each n,
n z n
+
the map w *â– Jl (w)dX (w) is bounded by xm(w) e L (P) , so these
n
converge to w + Jlft(w)dX^(w) a.s. and in l'(P). In particular,
E(J1ft(w)dXz(w)) exists. The proof for decreasing sequences is the
same.
H contains a semiring generating M. Let S be the semiring of half
open rectangles in the plane from before, and denote P = sf^R^.
Then U  (AxF, A e P, F e j} Is a semiring generating M. For a set
B e U, not only is E(1 dX ) defined, but we can compute it
â€˜ B z
explicitly. There are four types of sets in P (cf. Theorem 3.2.2):
1) A  C(s,t),(s',t')]: then E(il, ^.dX ) = E(1_/l.dX )
i Axr Z 1 r
FJ A Z
E(1 â€¢ (A (X))).
F A
2) A = (s,s'] x C 0,t' ]: then we get E(f1 dX ) =
â€˜ AxF z
E(1 (X, , .  X,
F (s ,t ) (S,t )
A â– = [0,3'] x (t,t']: we get E({lAxf,dXz)
E(1.,(X , ,:>)â€¢
F (s ,t ) (s ,t)
3)
115
4) A  [O,s'] x [O.t']: we get E(il dX ) = E(1â€žX. . .,).
1 AxF z F (s t )
(Note: See Theorem 3.2.2 and preceding example for computations of
the measure dX(w) on these rectangles.)
By the Monotone Cla33 Theorem, H contains o(U) = M, so E(fl dX )
â€˜ M z
is defined for all M e M. Set p (M) = E({ 1 dX )â€¢ Then pY: M + E
X ^2 M z x
+
is a oadditlve stochastic measure: ux is evidently additive. If
M * $, then for each w, fl (w)dX (w) + 0 by oaddltivity of the
n J M z
n
integral. Then E(/1M dX^) > 0 by Lebesgue; hence px Is oadditive.
n
Also, if M is evanescent, then M(w) Is empty Pa.s. => /lM(w)dXz(w)  0
a.s. > px(M) = E(/1MdXz> = 0.
Now, also satisfies ux(M)  Â£ UX(M), since U)((M)  =
E(/lHdxzÃ * E(/lMdXz) Â£ E(/lMdxz) = UX(M) (UX is the measure
associated with x by Theorem 4.1.4); hence ux has finite variation
uxl Â£ Px (since the variation is the smallest positive measure
bounding the norm). We shall prove this is an equality.
Each X^, being measurable, is almostseparably valued, so we can
find a common negligible set Nq outside of which Xz is separately
valued for z rational. By right continuity, X  lim X ,
z u
u+z
u rational
so for w i N^, X takes on values In a separable subspace
Eq C E. Let ZCE' be a separable subspace norming for Eq. Since
ux Â£ u  x  which is finite, we have yx << Uxj* By the extended
RadonNIkodym Theorem (Theorem 1.5.8) , there exists a stochastic
function H: R2xi2 â€¢* Z' (L(R,Z')) having the following properties:
116
1') H Is y,,measurable and h Â£ 1. In fact, Theorem 1.5.8
says that lÂ« is y xintegrable and that for ip z L1(yx)
we have Ji*iux = /HÂ¡ji^dy X  TaklnS +  1ft, A e M, we
obtain ] ux(A)  JH â€¢ 1AdyX " U j X^A ^1 hence lHl  1 on
A, so h E S 1 except on a yxnegligible set (on which we
modify H appropriately, say by setting H = 0).
2') is y. .integrable for every z z Z, and we have
 ^dyx for every M e M. In fact, taking
f * 1 in Theorem 1.5.8 (2), we get yxintegrable for
all z. Also, for z z 1 and M e M, we have, taking f = 1â€ž,
 Jdyi  x  , i.e., = /1 Mdy j x 
(since = 1h) = /Mdy,x..
3') yx(M)   H  du  x  for M e M. We showed this in proving
(1 ').
Now, taking M = [0,u] x A, A z F, in (2'), we deduce that
(11.2.3)
E(1 )  E(1 L ,d IXI )
A u A; [ 0, U J W 1 1 W
In fact, on the left hand side of (2'), we get =
 = = <Ã X dP,z> =
J ^2 M w AJ [0, u] w A u J A u
/AdP  E(1A), which is the left hand side of (4.2.3). As
for the right hand side, JMdy^ = E(JlHwdXy) (by Theorem
Ãœ.1.K, since  S hz < z) = E (1 aÃ[0iU]â€žd  X w) , which
is the right hand side of (it.2.3), thus proving the equality.
Now, since (4.2.3) holds for all A e F, there is a Pnegligible
set (depending on u and z) N(u,z)C n outside of which
= , d IXI . Since both sides of this equation are
u J [ 0, u ] v 1 1 v M
right continuous, there is a negligible set N9x)  N(u,z)
u rational
outside of which = , dIXI for all u e R2.
u â– '[O.u] v ' 'v +
Let S be a countable dense subset of Z and set N = ( N(z);
zeS
is negligible. Also, since X^ is integrable, there is a third
negligible set outside of which x^(w) < Â®. Let, now, w t
NQU Nl U N2 be fixed. The function X_(w) = X(w) is an EQvalued
2
function defined on R+, having bounded variation X(w)  = x#(w).
Then (Theorem 3.1.1) it determines a Stieltjes measure y on
X(w)
2 i i
B(R ) with finite variation y , . I. By Theorem 3.2.2, we have
+ a (w;
IÂ» Then by Theorem 1.5.7, there exists a function
lwX(w) I
1 X(w)
G : R â– * Zâ€™ such that
w +
1") l
Xw
a.e.
2") z> is y,x(w),integrable for every z e Z and
~ ^MdtllX(w) l fÂ°r M C B(R^* In faotâ€™
taking f  1 in 1.5.7(2), we obtain (as for 1.2')):
M
113
' ^dvJ X(w) I
^1MdyX(w)  = â€¢ÃMdyx(w) 
Taking M = [0,u], we obtain  , and
A { W ) U
fMdyX(w)] = i[0,U].dlXl.(w); henCe â€œ
/[0 uj^d X  . (w). Putting this together with what we had
2
earlier for , we now have, for Z z S, u e R ,
u +â€™
lCOtu]dlX].(w) = .z>  JC0,u].Z>.dx,(w).
1 2
There Is then a ^x(w)"neÂ®ligible set N (w,z)C R+ outside of which
we have = . In fact, the two integrals above forra
w
2
measures on B(R ). By taking differences, we have, for u
J(u u']dX(w) = J ^, dX(w), and these rectangles,
2
along with those [0,u] generate B(R ); hence 
â– + w
yX(w) faÂ¬
llow, the set N1(w)  l In'(w,z) Is y. .negligible, and for
zeS lX(w)l
u i N (w) we have = for all z e S; hence for all
u w
z e Z since S is dense in Z. Since Z is norming, we have
H (w) = G (u) for u i n'(w).
u w
Let A = {(u,w): H (w)I < l). A is then y^measurable (in fact,
A  AqI^I N with AQ e M, N pxnegligible. Then N is pxnegligible
so A is pxmeasurable). For each w, consider the section
A(w) = {uH (w) < 1 }; since for w i NQ\J N1 we have
K\ * lyx(w) 'ae and HU(W) = GwCu) uX(w) 'aeâ€™ we deduce that
A(w) = u: Hu(w) < 1} = {u: Gw(u) Â¡ < l) (a.e.) is Ux(w)"
negligible. Then P xCA)  E(/ j1A(w)(u)dIxIu
119
f 2\(w) (u)dlxlu(â€œ) * 0 p_as Then  H j = Ipi^.a.e.; hence by
R ' '
+
(3') we have ux(M) = JM H Jdu j x   J^l du  x   u j x  ( M) for M e M,
i.e., ux  = UX. The remainder of the theorem ($ e L1(px) lff
E(i 2l*ud X^) < Â») will be proved in the next theorem in more
generality. B
p
Proposition H.2.2. Let E,F be Banach spaces and V: R+xS2 â– + L(E,F)
2
be a process with integrable for every z e R+ and with raw
integrable variation V. If X is an Evalued measurable process,
then x e iff X is n, .almost separably valued and
E V  X 
E( f IX I dIVI ) < Â». In this case, E(i â€žX dV ) is defined, and
J _2 â– z * 1 1z l ? z z
pâ€ž(X) = E([x dV ).
V J z z
Proof. One way is easy; if X e Lg(jv), i.e., X e L^, (y  v ), then X is
Pvalmost separably valued (being measurable) and E(J ^ X J ^d  V 
R
+
< Â». In fact, if X e L^,(uv), then xÂ¡ 1 = p .y. ( x) =
le(vv)
E(J ?X ld V ) by Theorem Â¿4.1 .M (which holds for positive measurable
R
processes as well by monotone convergence), and this is finite by
assumption.
For the other implication, let X be an Evalued, measurable
process, Pyalmost separably valued and satisfying the condition
E(f 2Xzd]vz) < Â». Let (Xn) be a sequence of p. ,measurable step
processes such that X 1 â– * X
v i va.e. and xn < X for every n. Let
120
A be a uvnegligible set outside of which X is sepaably valued
and Xn  X pointwise; then y, .(A) = E(J 1 dv ) Â« 0 Â«>
I" I A z
I _1 (w)dv (w)  0 Pa.s. Denote the exceptional set by N. For
R 2
+
w i N, the section 1^(w) is dV^(w)negligible, so
xâ€(w) Â» X^(w) dv (w) almost everywhere, and x\w)  < Jx^Cw) .
Now, since E(Xzjdv ) < â€, there is a Pnegligible set n'' e F
such that for w t N1, JX (w) dV  (w) < Â®, l.e., X(w) Â¡ is
d V (w)integrable. Then for w t N N1 we have
Jxn(w) i X(w)  e L1(dV(w)) and xn(w) Â¡ â– * x(w)JdV_(w)a.e,
so by Lebesgue / Xn(w)[dv (w) *â– / x (w)d]v (w) for
RÂ¿ z z RÂ¿ z z
+ +
w t N UN1 and in particular f Xn(w)dV (w) > f _X (w)dV (w)
'2 Z z J 2 z z
H n
+ +
for w i N (J N1 .
Repeating the procedure, since the map w  J x (w)dv (w)
r2 z z
+
is Pintegrable, J x^(w) dV (w) Â£ Jx (w) dV (w) Pa.s. and
J Xz(w) dV (v) â– * /X (w) dV (w) Pa.s., we can apply Lebesgue
again and deduce that we also have convergence in L^P); in
particular, E(J Jx"dv J  E(J x Jdv ). Moreover, for each n
RÂ¿ z z RZ z z
we have ECl/x^dVj) < E(/x"dvJ) < ECj Xz d  V  J < Â»; hence
Jx^(w)dlMw) e LR(P) for each n. (Note: We must 3how this map is
measurable; this will come out of a later computation.) We already
showed that Jxn(w)dV (w) > Jx (w)dV (w) P*a.s. so by Lebesgue the
121
limit is Pintegrable and we have E(fxndV ) â€¢* E( fx dV ). (in
J z z J z z
particular E(/x dV ) is defined). Next, we show that x t L1_(p ):
for each n, Uv(xn) =. E( j x" d  V  Â¿ S E( J XzdV2) < , so by
Fatou we have pv(lim infxnj) < lim inf Pv(xn() < â€, in
particular lim inf fxn is u ,; integrable. But Â¡x = limXnÂ¡
a.e. so X is p, .integrable. Also, x  lim Xn is p measurable,
' ' n v
so X e L^,(uv). Moreover, pv(Xn) â€¢* uv(X) by Lebesgue again, since
xnI < X e L1 (UV ).
Finally, we show that, for each n, we have u,,(Xn)  E(ixndV ),
V J z z
so we get the desired equality by passing to limits. Being a step
k
process, we can write Xn = 1 1 X,, M. e M, X, e E. Then we have
1 = 1 Mi 1 1 1
uv(Xn) = Jf) = (Mj) = EXj(E(l^ dV^)) (by Theorem Â¿1.2.1)
 EE(x. fl dV )  EE( Ã1 x dV ) (by Theorem H.l .3) = E([(I1 , x,)dV )
i* z J Mj 1 z J Mj i z
* EfJx^dV^) (and in particular the map Jx^(w)dVz(w) is P
measurable). Letting n + Â», we obtain p,,(X) = E(fx dV ), and the
V 1 z z
theorem is completely proved. I
Remarks.
1) By taking E = R in the statement, we have the following: If
X is any scalarvalued measurable process, then X e l'Cpy) iff
122
E(f X zdV  ) < (X is automatically separably valued.) Then
E(xzdVz) is defined, and Uy(X) = Etjx^dV^). Finally, equality
(H.1.2) is proved the same way as (t.1.1), by taking step processes
and passing to limits.
2) The correspondence is not onetoone: as we shall see In the
next section, a stochastic measure with values in L(E,F) is generated
by a stochastic function{not necessarily measurable) with values in a
subspace of L(E,F").
We can also generate stochastic measures from stochastic
functions (not necessarily measurable) with raw integrable variation,
as the next theorem shows.
Theorem t.2.3. Let E,F be two Banach spaces and Z C F' a subspace
2
norming for F. Let B:R+xÂ£l + L(E,F) be a rightcontinuous stochastic
function satisfying the following conditions:
i) B has raw integrable variation b.
ii) For every x e E and z e Z, is a realvalued process
(measurable!) with raw integrable variation .
Then there exists a stochastic measure m: M â– * L(E,Z') with finite
variation m satisfying the following conditions:
1) If X is an Evalued process and If X is y, .integrable, then
lBl
X e Lâ€ž(m), the integral E(<Ã X (w)dB (w),z>) is defined for every
& U U
Rt
Z E Z,
= E(< fx dB ,z>),
1 u u
123
and
M;x) < E(/XudBu).
2) If, in addition, Bx is separably valued for every x e E and
if X is y i iintegrable, then the integral E( f â€žX dB ) is defined, and
Ã31 jr2 u u
+
m(X)  E(f X dB ).
V u u
+
3) The measure m has values in L(E,F) in each of the following
cases:
a) F  Z\
b) For every x t E and v e R^, the convex equilibrated
(balanced) cover of the set (3 (w)x: w e ill is
v
relatively o(F,Z)compact in F.
o
c) For every x e E and v e R , the function 3 x is F
+ v
measurable and almost separably valued; in particular,
this is the case if F is separable.
Proof. Let pB be the measure generated by 13] via Theorem i).1 .Â¿l.
For every x e E and z e Z the variation of the process <3x,z>
satisfies v < b Â¡xz) (cf. proof of Prop. 4.1 .3).
Let z be the stochastic measure generated by :
mv ,(M)  E(J 1 d) for M E M. The mapping (x,z) â€¢* m (M)
X,Z ^ X'Z
+
is linear in each argument: in fact, m (M) =
*i X2Â»z
E(i 21Md) â€œ E(JlMd) = E(/lMd(+<3x2,z>))
R
 E(/i Md + /l Md)  E(/lMd)+E(/lMd) =
m (M) + m (M). The computation for z Is completely analogous.
xj Â» 2 *2* ^
Also, we have mx z(M) = E(/ 1^d)Â¡ Â£ E(\j 1 d) S
1 FT R^ V
+ +
E) 2 E
R R
+ +
 ixl' llzlâ€˜dB (M) . so mx z(M)  < x J zp 0  (M). Then for given
M z M, the map (x,z) *â– m (M) Is continuous, bilinear. Then there
X , z
is a continuous linear map m(M) z L(E,Z') satisfying
= m (M) = E(/ .1 d).
X t z n v
(More precisely, any continuous, bilinear function f(x,z): ExZ â– + R
is continuous and linear in each component. Then the map x â€¢+ f(x,*)
is a continuous linear map from E into Z'. In our situation, we
have m(M)x = mx ^(M), so = mx Z(H), i.e., for M t M, we
have ra(M) z L(E,Z').) We also have m(M) I Â£ y, ,(M), since
lBl
m(M)  = sup m(M)x  sup K .(M>l  sup ( sup [m (M) J) Â£
:<1
JxÂ£1
x,
IS1
x,z
sup ( sup SxIzui(M)) = u, ,(M). This gives us a map
[x<1 zÂ£1 'Bl
m: M * L(E,Z'). We now verify that m is oadditive and has finite
variation m (in particular m Â£ UB):
First of all, m is additive. Let M,N z M be disjoint; we show
m(Ml^/N) â– = m(M) + m(N) , i.e., that m(MVJN)x = m(M)x + m(N)x for all
125
x e E. This amounts to showing that =
for all x e E, z e Z. Now, = E(f 1 d)
1 HUN v
+
' E(J 2(lM+1N)d) â€™ E(l 21Md + / 21Nd) â€™
R + Rt R+.
E(/ 1 d) + EC/ ?1 d) = + =
R R
+ +
, which shows that m is additivie. If, now,
A + 0, m(A ) + 0 since Im(A )1 Â£ u,â€ž,(A ) and the latter is
n * n â€™ n "  B  n
oadditlve. Then m is oadditive as well.
As for the variation, y, , is a bounded, positive measure
!BI
satisfying m Â£ uB; hence m Â£ pg since the variation is the
smallest positive measure dominating the norm. In particular, m has
finite variation.
Now we prove assertions (1)(3).
Ad (1): Let X be an Evalued process. From the inequality
hi  PB it follows that if X e Lg(uB), then X e Lg(m)
(since any sequence of step functions Cauchy in l].ui , is then also
E  B 
Cauchy in Lg(m)Â¡ hence X e Lg(m) since X is Hmeasurable, and
M(*l>  UB(XJ) = e(J 2!xv rd B  v) , which is the second part of
(1).
The first part of (1) is satisfied for any Mmeasurable step
n
process X * I 1 x., M e M disjoint, x. e E. In fact, for
i1 i 1 1
n n
z e Z, we have = = < I m(M.)x.,z>
ii Mi 1 i=i 1 1
126
n n r n ,
 I <â– (Â«.)* ,z> = l E( 1 d) = I E() (by
11 1 1 i1 R2 Mi V 1 i>1 R2 M1 V 1
+ +
Prop. 11.1.3)  E( I )  E() 
11 v 1 J v i
E() (again by lJ.1.3)  E() =
E(). Now, let X e Lg(yg) and let Xn be a sequence of
measurable step functions such that Xn â– * X uBa.e. and Xn  < x
everywhere. Let A be a uBnegllglble set outside of which X is
separably valued, Jx dBv < Â» (more precisely, for (w,u) i A,
^[0,u]lXvCw)!dlBlv(w) < slnce E(/ 2lxvIdlBlv> <
R
+
J 2 lxv<Â«) ld [B1 v(w) <  Pa.s.; hence ir0>u] lx./w) dB y(w) < Â»
Uga.e. since it is a Pmeasure), and X â€¢> X. There then exists a
Pnegligible set N e F such that for w i N, the section A(w) is
d 131 _(w)negligible (in fact, E(J1A (w)d(3y(w)) = Pb(a5 â€ 0 >
/ 21 A(w)d I9 I â€¢ ^ â€ 0 p'a's*)> so for w t N we have dB (w)a.e.:
R
i) X (w) is separably valued
11) X%) < x,(w)
HI) X%) â€¢Â» X.Cw).
Since  x (w) d B (w) < Â® for w Â¿ N, X (w) is d131 (w)
R V
+
integrable, and x"(w) â™¦ X.(w) in LÂ¿(dB^(w)), so by Lebesgue
127
xâ€(w)dBy(w) * Jx^(w)dB (w). Then, by continuity, for w t N, z e Z,
we have <â€¢ j moreover,
R2 V v R2 v
 < IzI/x^(w)dBv(w) < z.fx"(w)d3v(w) <
fiz /Xy(w) d 3  ^(w), Now, the function w > J x (w) dB (w) is P
R
+â–
integrable, so by Lebesgue is Pintegrable and
Ft v v
+
E( *â€¢ E() for all z e Z. For each
n, Â» E() as we saw above. Finally, X is
Mmeasurable by assumption, and [x e L1(pB) C L1( m)Â¡ hence
X e Lg(m), and m(Xn)  m(X) Â£ ]m(xn  X) * 0 by Lebesgue (since
Xn <  X , Xn â€¢* X ma.e.), i.e., m(Xn) *â€¢ m(X). Then Â»
for all z Â£ Z. Passing to limits, we obtain =
E() for all z e Z, which completes the proof of (1).
Ad (2): Suppose, now, that Bx is separably valued for every
x e E. Let X e l!(ui ,), let Xn be step processes convering to
e, J 31
X UBa.e. with xn  < x for ail n. We shall show fir3t of all
that the map w Â» J 2Xy(w)dBy(w) is integrable for all n; write
n
T 1^ x^, Â£ M disjoint, x^ Â£ E. For each i, Bx^ is separably
valued. Also, by (ii), is measurable for z e Z. Since Z is
norming, Bx^ is weakly measurable, so Bx^, being separably valued, is
128
strongly measurable, with integrable variation (in fact, 13x.] S
x. by Prop. 11.1.3). By Theorem it.2.1, E(J 1 d(Bx.)) exists.
1 RZ Mi 1
+
E(J ?1M d(Bx )) = E E(/ (1 x ) d3 ) (by Prop. it.1.3)
R I i1 Rz i 1 u J
+ +
E( E jl x.dB ) = E(/( E 1 x,)d3 ) = E(ixndB ) exists. We proved
, ,1 M l u J , , M, i u  u u
n
Then E
1 = 1 Rf "i
n
i=1 i
1 = 1 i
in (1) that JXn(w)dB (w) â– + ix (w)dB (w) Pa.s. Moreover, for each n,
/X%)d3u(w) Ã© j xâ€(w) dB u(w) Â£ JX (w) dB (w). By assumption,
the latter is Pintegrable, so by Lebesgue Jxu(w)dBu(w) is P
integrable, and we have E(Jx^(w)dBu(w)) *â– E(Jx^CwJdB^tw)). We have
from before that m(Xn) > m(X). It remains to prove that
m(Xn) = E(JxndB ) for all n. For all z t Z, we have =
= = E = Em (M.) =
j_1 Hj i 1 i i i x^z l
EE(/lM d) = EE() = 5ZECz>)
E(Z) = EC) = E(
, z>)
. (Note: We can now do this last step since
Jx^(w)dBu(w) is Pintegrable; it was not in part (1)!) Both m(Xn)
and E( XndB ) are Z'valued, so this means that m(Xn) =
1 u u
E(J 2XÂ«d3u) for all n. Passing to the limit, we obtain
m(X) = E(/ 2XudBu) ; in particular, the double integral on the right is
defined.
129
Ad (3):
(a) is trivial,
(b): Let x e E, v e R+. Since the set C  coBv(w)x: weÃj}
(balanced closed convex hull) is oCF,Z)compact, the natural embedding
of C in Z*, the algebraic dual of Z, is o(Z*,Z)compact (see Dunford and
Schwartz [8]). There is then a family (z.) of elements of Z such that
i ÃeI
C  PllytZ*:   Â£ if (any closed convex set is an intersection
isl
of halfplanes; we can use balls since C is equilibrated). Then we
have  Â£ 1 for all i e I, w e Q. Let M = [0,u] x A,
A e F; we have   = E( fl r _ , ,d) 
1 i " 1 1 [0,uJxA v * i "
E(1A)I Â£ E(1 J) Â£ 1; hence m([0,u]xA)x e r,
i.e., m([D,u]xA) e L(E,F). By taking differences, we have
m((u,u']xA) e L(E,F), and also finite disjoint unions of such sets.
We shall use the monotone class theorem to prove that m(M) e L(E,F)
for all M e M.
Let M * (m e M: m(M) e L(E,F)]. We show first that is a
monotone class: Let M e M , M + M. Then m(M )x e FC Z'. We have
n n n
Jm(Mn)x  m(M)x Â£  m(H)[x[ 0 by oadditivity of m. Hence
m(M^)x + m(M)x in the metric topology of Z'. Since F is closed in
Z' for the metric topology, m(M)x e F as well, i.e., m(M) e L(E,F).
The proof for + M is exactly the same.
Now, let C be the algebra generated by sets of the form
(u,u']xA, A Â£ F. C consists of finite unions and complements of such
sets. We have shown that if M is a finite union of such sets, then
130
c p
m(M) e L(E,F). As for complements, m(M ) = m(R+Xii)  m(M), and
2
m(R+xQ)x = lim m([(0,0), (n,n)]xii)x e F by closure of F in Z' as
before. Thus, if m(M) e L(E,F), then m(MC) e L(E,F). Then CC M , C
is an algebra, so M = a(C)Ch>/ by the monotone class theorem, l.e., m
takes values in L(E,F).
p
c) Suppose that for x e E, v e R , the function 3 x is F
+ v
measurable and almost separably valued (in particular, if F is
separable, then B^x is separably valued and weakly measurable by (ii);
hence B^x is Fmeasurable). Then for every A e F, x e E, the function
I^B^x is integrable; in fact, B^x is Fmeasurable, by assumption
and we have Bvx Â£ 3VIIX E l'cp). We also have, as before,
= E(/lr n .d) = E() =
; [0,v]xA â€¢ â€˜ [0,v]xA
= (again, we can move the
1A [0,v] â€¢ A v
expectation inside since l^B^x is integrable). Since this holds for
all z e Z, we conclude m([0,v]xA)x â– EO^B^x) e F. Then
m([0,v]xA) e L(E,F), and we conclude by the same monotone class
argument as in (b). I
Remarks.
1) This theorem shows that if B has values in L(E,F), then m
B
has values in a subspace of L(E,F"). Moreover, we do not have in
general Im  = p, ,. Later we will establish some conditions
B  B 
sufficient for equality.
2) The correspondence B â– * m is not injective. For an example
involving measures associated with functions, see Dinculeanu [6,
p. 273].
131
iJ.3 VectorValued Stochastic Functions Associated
With Measures
In this section we consider the converse; starting with a
stochastic measure m with finite variation, we will find a stochastic
function B with integrable variation such that m Is associated with B
in the sense of Theorem H.2.3. The precise result is the following:
Theorem 4.3.1. Let E,F be two Banach spaces and Z Cl F' a subspace
normlng fo" F. Let m: M > L(E,F) be a stochastic measure with finite
variation m. Then there exists a right continuous stochastic
function B: R^xi! >â– L(E,Z') satisfying:
i) B has raw integrable variation b.
ii) For every x c E and z e Z, is a realvalued raw
process with integrable variation .
Moreover, we having the following:
1) If X is an Evalued measurable process we have X e lVih)
E
if and only if X e L^Cp. .), In this case the integral
E 13 ]
E() Is defined for every z e Z,
R
+
= E(), and
;02 u u
n
+
IH<X)  E(/ 2XudBu), i.e., m[ = pB.
2) If F is separable (or more generally if Bx is separably
valued for every x e E), then Bx is measurable for every x e E.
If B Is separably valued, then B is measurable.
132
3) We can choose B with values in L(E,F) in each of the
following cases;
a) F is the dual of a Banach space H and we choose Z = H;
hence F = Z'.
b) For every x e E, the convex equilibrated cover of the set
(Jxdm: $ simple process, J$dm < 1 ) is relatively o(F,Z)
compact in F.
c) E is separable and F has the RadonNikodym property (we
say F e RNP); in this case B can be chosen such that Bx is measurable
and separably valued for every x e E, hence
m(X) = E(JxudBu) for X e L^m).
d) The range of m is contained in a subspace G L(E,F)
having the RNP: in this case B can be chosen measurable, with
separable range contained in G; hence
m() = E(Jif^dB^) for <> e L1(m).
4) If p is a lifting of P, we can choose B uniquely up to an
evanescent set, such that p[B ] = B for every v e R^ (see Definition
V V +
1.5.5(b)).
Proof. Let V be the integrable increasing raw process associated with
!m via Theorem H.1.4:
Im!(M) = E(f 1 dV ) for H Â£ M.
11 JH u
133
2 2
Denote the rectangle [0,z] by R^; for z e R~ set ra (A) = m(RzxA)
for A e F. We verify that mZ: F â™¦ L(E,F) Is a oaddltlve measure
with finite variation mZ, and that mZ is absolutely Pcontinuous:
i) m_ is qadditlve: First of all, mz is additive. Let
A,B e F, disjoint. Then R^xA and R^xS are disjoint, so we have
mZ(A^JB) = m (R x(A{J 3)) = m((R xA)^,'(R x3)) = m(R xA) + m(R x3)
Z 2 Z Z Z
= mZ(A) + mZ(B), Now, let (A ) t F, A + 0. Then (R xA )+(R x 0)
n n z n z
= 0, so lim m (A ) = lim m(R xA )  ra(0) = 0. so m is indeed
n z n
n n
oadditive.
2
ii) m has finite variation: we show in particular that
m2 < [m Z, where mZ(A) = m(RzxA). Let A e F, and let (A ^),
n
1  1,...,n be disjoint sets from F with Ã^/A C A. We have, since
i1 1
n n n
Â»x(U*,)  U(S *A.)CR xA, I mZ(A ) I
2 1=1 1 iVi 2 1 2 1=1 1
n
n
Â£ m(R xA.)  <
i = 1 2
E m(R^xA.) Â£ m(R^xA) = mâ€œ(A). Taking supremum, we obtain
mZ(A) Â£ mZ(A).
ill) rr,Z << P: In fact, we have mZ(A)  = m(RzxA) 
Â£ ra(R^xA) = E(j 21R xAdV ) = E(1 V ); hence ]mz Â« P, so
R+ 2
2
m << P as well.
Applying the Extended Radon Nlkodym Theorem (1.5.8), we get, for
2 0
each z e R+, a function : n â™¦ L(E,Z') satisfying:
o 1
1) Bz is Pintegrable, and for iji e L (m ), we have
id m
JBj*dP.
134
1 7
2) is Plntegrable for all f e L^(m ) and zQ Â£ Z,
and = idP.
â€˜ 0 ' z 0
so "
3) If p is a lifting of L (P), we can choose (Bz) uniquely
0 0
Pa.s. such that pCJ  Bz> i.e., for all A Â£ F, x e E, zQ e Z,
0 Â«> o o
we have 1ft e L (P), and p(1ft) = 1p(A)â€˜ Ifâ€™
addition, there exists a>0 such that mz Â£ aP, then we can choose
o 0 0 0 m
B uniquely everywhere such that p(B ) = B , i.e., Â£ L (P)
z z z z 0
0 0
for all x e E, z e Z, and p() = for all x,z .
v Z U Z U (J
4) If one of the conditions in 3(a) or 3(b) is satisfied, then
0
B takes values in L(E,F).
z
Now, in particular, taking i> = 1 , A e F in (1 ) we obtain
1 ') mZ J (A)  EOjBj) for A e F.
Also, taking first f â– x, x e E, and then f = xlfl, A e F, we get
0
2') is integrable for x e E, zQ e Z, and
= = /dP = E(1A), for
A e F, x e E, z0 e Z.
0
(Notice also that from (1'), if B is bounded, the condition
z
mZ Â£ aP in (3) is satisfied.)
From (1') and the inequality mz Â£ mz we obtain
flBÂ°) = mZ(A) Â£ mZ(A)  m (IMcA)  E(/lR dVu> = E(1 V ),
z
D 0
i.e., E(1 B 1) Â£ E(1 V ) for all A Â£ F; hence B  Â£ V Pa.s.
A Z A Z Z Z
2
Let z = (s,t), z' = (s',t') be points in R , z < z'. Denote by
D , the set R , \ R , and by R
zz z z zz
in
the rectangle t (s,t), (s', t') ]
135
We have m(D xA)  m((R \ R )xA) = ra((R ,xA) \ (R xA)) =
Â¿Â¿ z z z z
m(R ,xA)  m(R xA) = mZ (A)  raZ(A). Likewise, ra(R ,xA) =
, ZZ
m (A)  m ''(A)  mSt (A) + mSt(A). Then for x e E, zQ e Z,
zy z 7 '
we have <(ra m )(A)x,zQ> =  =
0 0 0 0
E(1a)  E(1a). The same
computation gives <(mS  mS t  mst + mSt)(A)x,zQ> =
o o
E(1 <(A (BÂ°))x,z >). Now, since p[B = B etc., we have
zz' u s c 3 t
0 0 0 0 0 o
PÃX'  B ] = B ,  B , and p[Â¿ (B )] = AD (B ). In fact (we
Z Z Z Z n , n ,
ZZ ZZ
give the proof for the first; the second is the same), for A e F,
x e E, zQ e Z, we have
<(B,'  VxV\  E Lâ€œ(p)
since each term is. Also, p(<(B ,  B )x,zâ€ž>1â€ž) =â– p(1 
Z z U A z 0 A
o o oo
1A) " P(1 )  p(1A = <3 ,x,z.>1
Wâ€™pU) = <(V ~ Bz)xV 1 p(A) â€™ SÂ° p[Bz' â€˜ Bz] = Bz' ' Bzâ€™
0
and the same for ar (B ). Then, by Proposition 1.5.6, both
zz'
I 0 0 0 0 0
3Z* â€œ Bz and ar (B ). are Pmeasurable. Also, B ,  B 
zz' z z
oo o o
â€¢ + B I S V â€ž + V Â£ 2V hence B ,  B I is Pintegrable;
z z z z z z z
o o
similarly, Â¿ (B )  < Â¿4V hence Iad (B ) is Pintegrable as
* Z * H ,
zz zz
o o
well. Also, by properties of liftings, <(B ,  B )x,z> and
z z
o
<(Ar (B ))x,zq> are measurable for x E g, z e Z (see property 2
136
after Defn. 1.5.5). By the "converse" of the generalized Radon
Nikodym Theorem (Theorem 1.5.9), there exist measures
m^: F * L(E.Z') and mR: F * L(E,Z') (the measures have the same
o
values as the function B since Z' is a dual (cf. part 3(a) of the
statement of this theorem) with finite variation mD and mR
such that:
0 0
1) = E(1 <(B .  B )x,z_>) for A e F, x e E,
D U A z z 0
zQ e Z (by taking f = x1fl in 1.5.9), and =
o
E(1A<(AR (B ))x,zQ>) likewise. Also,
zz'
ii) mD(A) = E(1aBÂ°,  Bj), and mR(A) = E(1AAR (BÂ°)Â¡)
ZZ '
for A c F (we take ty = in 1.5.9).
0 0
From (i) we have =â– E(1, <(B ,  B )x,zâ€ž>) 
u u A z z 0
<(m2  mZ)(A)x,z > from earlier. Likewise, =
U K U
S't ' s't
<(m  m
s t ^ s t
m + m )(A)x,Zq>. Both these hold for all A,
x, Zn so we have mn(A)x = (mz  mZ)(A)x, and mR(A)x
(mâ€˜
0
s't'
s't
D
st'
+ mst)(A)x for all A, x; hence m^ = mZ  m2
and m * m
n
S't'
s't St'
m  m
st
(and in particular m^, mR
have values in L(E,F)),
By (ii), we have mZ  m21(A) = mD(A) = E(1Bz,  B^J),
q ' q *t" q t * q j 0
and similarly m  m  m + m (A) = E(1aar (B )) for
zz'
A e F. On the other hand, we have (m  m )(A) = m(Dzz,xA)j <
m(Dzz,xA); hence mZ  mZ(A) < m(D ,xA) = mZ (A)  mZ(A)
 E(1AVz'}  e(1aV  E(VVZ'V2^ hence e(1aÂ¡bÂ¡'  3J> Ã¡
>37
E( 1 (V Ã )) for all A e F, so 3 ,  B I < V ,  V Pa.s. for
A z z 1 z zâ€˜zz
each z < z'. The same computations for the rectangle yield
o
ar (3 )[ Ã (V) Pa.s. If we take z,z' with rational
zz' zz'
0
coordinates, we can find a common negligible set and modify B
on it to get the Inequalities everywhere for all z,z' rational.
Next, let z be fixed. We show that for any sequence r + z, r
n n
0
rational, the sequence (B (w)) is Cauchy for all w. In fact, the
n
sequence (V (w)) is Cauchy for all w since V is right continuous,
n
i.e., for any e>0, there exists n^ such that n,m > implies
It 0 0
V (w)  V (w) < e. Then, for n,m Â£ n , we have B (w)  B (w)
J n m n r m
0
Â£ V (w)  V (w) < e. Thus, for any w, the sequence (B (w)) is
n m n
o
Cauchy in L(E,Z') complete, so lim B (w) exists.
r
n n
Now, let (rnK(sn) be two sequences of rationale decreasing to
z. We can construct a sequence (v^) decreasing to z, containing
0
subsequences of both (r ) and (s ), Then lim B (w) exists; moreover,
n n v.
J J
0 0
since lim B (w) and lim B (w) exist, and subsequences of both are
n rn n 3n
O
contained in (3 (w)), all three limits are equal. In particular,
VJ
0 0
lim B (w) = lim B (w), so we get the same limit for any sequence of
n rn n Sn
2
rationals decreasing to z. Then, for every w e fi, z â‚¬ R ,
B (w) = lim B (w) exists. The stochastic function B thus
z , r
r + z
r rational
defined is right continuous.
138
In fact, let e>0: there exists a neighborhood to the right of z
so that if r is rational and lies inside the neighborhood, then
b^(w)  B^(w)  <  . For any z' in this neighborhood, there exists a
similar neighborhood for it, and the intersection of these has
nonempty interior. Let r be in their intersection, r rational. We
have Bz(w)  BzÂ«(w) = Bz(w)  BÂ°(w) . BÂ°(w)  B7.(w)
B (w)  B (w) + B (w)  B ,(w)  < 
r r'
e Â£
z r
continuous.
Thus 3 is right
Some more properties of B are the following:
Ð°) For w
o o
this, let q,r be rational, with q>z, r>w. Then B  3  â–
o o
B  B + B B +B  B I < V  V . Letting q+z, we get
â– q z z w w r1 q r o
o
Bz  Bw + Bw  B^[ Â£ Vz  by definition of B and right continuity
of V. Letting r+w likewise, we get 3  B I Â£ V  V . We
1 z wâ€˜ z w
similarly have Â¡A (B) Â£ A (V).
R , R t
zz zz
o 2
Ð±) For each zt B = B a.s.: In fact, for z e R , r rational,
o o
r>z we have Ib  B I $ V  V a.s. (in fact there Is a common
1 r z1 r z
negligible set outside of which this holds for all r>z) letting r + z,
D 0 0
we have b  B  = limB  B  < lim(V  V )  0 a.s., i.e.,
r + z r + z
0
B = B Pa.s.
z z
Next, we show that B has raw integrable variation 3. Since
0 DO
B  B Pa.s., B,B = B ,  B a,s.f and A_ (3) =
zz z zz z R ,
zz
o
A (B ) a.s.; hence p[B ,  B ] = 3'  B , and n[A (B)l
z z z zâ€™ R /
139
 Â¿ (B) (property (5) following Definition 1.5.5), so by 1.5.7,
zz'
Bz,  BzÂ¡ and ar (B)J are measurable; hence the finite sums we
zz'
use to compute Va^ fl] (B(. _ Q)) , Var [Q> t] (B(()>.,) , and
iar.,â€ž , ,.,(B) are also measurable.
L(0,0),(s,t)J
Moreover, since B is right continuous, we can compute the
variation using partitions consisting of rational points; the first
two terms from the onedimensional result, the third by Proposition
2.2.5. Each of these limits can then be taken along a sequence, so
Var[o,s](B(.fo))â€˜ Var[0,t](B(0,)) * Var[(0,0),(s,t)]
measurable; hence B
1 s,t
(0,0)1
Var[0.s](B(..0)) +
Var[0,t]B(0,) + Var[(0,0),(s,t)](B) 13 measurable, I.e., b is a raw
process, evidently Increasing.
We show now that B has integrable variation. We shall show, in
fact, that B Â£ V. We proceed with each term of B separately:
1) Slnce IB(0,0)I  v(0,0) we have lB(0,0)l  v(0,0) as
2) Let o: 0  s. < s, < ... < s = s be a partition of [0,s],
U 1 m
m1
We have L B
i = 0
 B,
m1
I S 1 (v,
 V,
(si + 1,Â°) (s^ ,0) 1 â– 1
) =
iV(sr0) V(0,0)) + (V(s2,0) ' V(s1,0)) + â– " + (V(s,0) " V(sn]_1,0))
 V. .  V,â€ž Pa.s. Now, let (o ) be a sequence of partl
is ,Ui (U,UJ n neN
tlons of [0,s] such that Varrâ€ž ,(B, â€ž.) = lim Varr _ _(B, â€ž,;a ).
10, sj (,0) L0,s] (,0) n
n
Each term is dominated by V,  V.â€ž a.s., and V,  V,â€ž
(s,0) (0,0) (s,0) (0,0)
does not depend on n, so taking limits we have Varr ,(B, ) Â£
[Q,SJ 1 * , O J
v  v
(s,0) (0,0)
a.s.
3) The same argument gives Var[0)t](B(Qj  V(00)
*0 Let oxt be a grill on [(0,0),(s,t)]: we have Z i (B)
V(w)
Reoxt
 R^ AR(V) = V(s,t) â€˜ V(0,t) â€˜ V(s,0) + V(0,0)'
To see this last equality, consider for each w the measure m,
2
on B(R+) associated with V(w). Consider the halfopen rectangles
R associated with the closed rectangles R of oxn. Then
â– .VM(UÂ¡) â€¢ mv(u)U(0,0Ms,t>])  v(>it)  Y(0it)
 V
(s,0) (0,0)
of grills as before, we have Var
[(0,0),(s,t)]!B) Â£ V(s,t) ' V(0,t)
" V(s + V^0 ^ a.s. Adding up (1)(H), we obtain B^g ^ 
^(O.O)1 + Var[0,s]iB(,0)) + Var[0,t](B(0,)) + Var[(0,0),(s,t)]CB)
â– V(0,0) + (V(s,D) " V(0,0)) * (v(0,t) â€œ v(0,0)) +
â– V(0,t) + "(O.O)5  v(s,t> aS Thu9' for Â£ R' l3l(s,t) 
V(a a.s. Since  B [ is increasing, and 131 ^ ^ ^ a.s., b
is finite outside an evanescent set*; hence b Is right continuous
outside an evanescent set. Then, since both b and V are right
continuous, b Â£ V outside an evanescent set. In particular,
Bb Â£ V_, so B has integrable variation, and this completes the proof
of (1).
* More precisely, for example is a negligible set N such that
n
w i N => 13], (w) < V .(w); hence IBI Â¿ V,
n 1 (n,n) (n,n) 1 1 (n,n)
outside N^. Then NQ = is negligible and B[ < â– > outside
this set. n
mi
Now for (ii). For x e E, z. e Z we have =
0 z 0 z 0
a.s., so by (2') is integrable; moreover, is right
continuous since 3 is, and has integrable variation (we showed in the
proof of 4.1.3 that (w)] < B(w) x  ]z ). Also, for each  