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Vector measures and stochastic integration

Material Information

Title:
Vector measures and stochastic integration
Creator:
Witte, Franklin Pierce, 1942- ( Dissertant )
Brooks, James K. ( Thesis advisor )
Varma, Arun K. ( Reviewer )
Dufty, James W. ( Reviewer )
Bedmarck, A. R. ( Degree grantor )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1972
Language:
English
Physical Description:
v, 108 leaves. : 28 cm.

Subjects

Subjects / Keywords:
Algebra ( jstor )
Banach space ( jstor )
Indefinite integrals ( jstor )
Martingales ( jstor )
Mathematical integrals ( jstor )
Mathematical vectors ( jstor )
Mathematics ( jstor )
Measure theory ( jstor )
Perceptron convergence procedure ( jstor )
Scalars ( jstor )
Dissertations, Academic -- Mathematics -- UF
Mathematics thesis Ph. D
Measure theory ( lcsh )
Stochastic integrals ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
This dissertation investigates the stochastic integration of scalar-valued functions from the point of view of vector measure and integration theory. We make a detailed study of abstract integration theory in Chapters I and II, and then apply our results in Chapter III to show that certain kinds of stochastic integrals, previously defined by other means, are special cases of the general theory. In carrying out this program we prove extended forms of the classical convergence theorems for integrals. We also establish a generalization of the standard extension theorem for scalar measures generated by a left continuous function of bounded variation. The special case of measures in Hilbert space is discussed, and a corrected form of a theorem of Cramer is proved. In Chapter III we show that certain sample path integrals, the Wiener-Doob integral, and a general martingale integral are included in the abstract integration theory. We establish a general existence theorem for stochastic integrals with respect to a martingale in L , 1 < p < infinity>.
Thesis:
Thesis -- University of Florida.
Bibliography:
Bibliography: leaves 105-107.
Original Version:
Typescript.
General Note:
Vita.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit 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:
022670731 ( AlephBibNum )
13989845 ( OCLC )
ADA5262 ( NOTIS )

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To Do



Whose patience and encouragement made this possible.















ACKNOWLEDGEMENTS


I would like to thank my wife Dot for her under-

standing and encouragement throughout the past four years.

I am indebted to each member of my conunittee; special

thanks are due to Dr. James K. Brooks, who directed my

research and guided my study of abstract analysis, and

to Dr. Zoran R. Pop-Stojanovic, who guided my study of

probability theory. Finally, I would like to thank

Jean Sheffield for her excellent typing.


iii
















TABLE OF CONTENTS


Acknowledgements

Abstract

Introduction


Chapter

I. Abstract Integration

II. Stieltjes Measures and Integrals

III. Stochastic Integration

References

Biographical Sketch


Page
iii


105

108













Abstract of Dissertation Presented to the Gr-iaduate
Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy

VECTOR MEASURES AND STOCHASTIC INTEGRATION

By

Franklin P. Witte

December, 1972

Chairman: J. K. Brooks
Major Department: Department of Mathematics

This dissertation investigates the stochastic integration

of scalar-valued functions from the point of viev. of vector

measure and integration theory. We make a detailed study of

abstract integration theory in Chapters I and II, and then

apply our results in Chapter III to show that certain kinds

of stochastic integrals, previously defined by other means,

are special cases of the general theory. In carrying out

this program we prove extended forms of the classical con-

vergence theorems for integrals. We also establish a gener-

alization of the standard extension theorem for scalar

measures generated by a left continuous function of bounded

variation. The special case of measures in Hilbert space is

discussed, and a corrected form of a theorem of Cramer is

proved. In Chapter III we show that certain sample path in-

tegrals, the Wiener-Doob integral, and a general martingale

integral are included in the abstract integration theory.

We establish a general existence theorem for stochastic in-

tegrals with respect to a martingale in Lp, 1 < p < m.

v













INTRODUCTION

This dissertation concerns stochastic integration, a

topic in the theory of stochastic processes. A stochastic

process (random function) is a function from a linear in-

terval T into an L space over a probability measure space.

In its broadest terms, stochastic integration deals with

linear transformations from a class of functions x to a class

of stochastic process fxdz, which depend on a fixed process z.

The functions x which are transformed may be sure (scalar-

valued) functions, or more generally, random functions. Since

random functions take their values in an L space they are

vector-valued functions. Hence we are in the framework of the

general theory of bilinear integration developed by Bartle [1].

The vector-valued function z generates a finitely additive

measure on an appropriate ring or algebra of sets. When the

integrand x is a sure function we are concerned with the in-

tegration of scalar functions by vector measures. When the

integrand is another random function we have the full gener-

ality of the Bartle Theory, where both the integrand and the

measure take values in Banach spaces X and Z respectively,

and there is a continuous bilinear multiplication from XxZ

into another Banach space Y, in which the integral takes its

values.





2

Until recently the field of stochastic integration

appeared to be in much the same state as the area of real

function theory was before the systematic introduction of

measure theory. Several relatively unrelated kinds of

stochastic integrals can be found throughout the literature

on stochastic processes and their applications. Most of

these integrals are defined as limits of approximating sums

of the Riemann-Stieltjes-type. This approach, however, is

not well-suited to stochastic integration in general, and

certain difficulties may arise. With the exception of some

work by Cramer [10], the measure properties inherent in the

stochastic integration scheme have not been exploited.

There is currently a great deal of interest in unifying

the field of stochastic integration using vector measure and

integration theory. E. J. McShane, who has been investigating

stochastic integration for several years, is presently writing

a book which should contribute to this unification. The

recent Symposium on Vector and Operator-valued Measures and

Applications (Salt Lake City, August, 1972) was specifically

concerned with stochastic integration, and attracted many of

the known experts in both the fields of measure and integration

theory (Brooks, Dinculeanu, Ionescu Tulcea, Kelley, and

Robertson) and stochastic processes and stochastic integration

(Chatterji, Ito, Masani, and McShane). A very recent un-

published paper by Metivier [23], presented at the Symposium,

deals with the problem of incorporating a rather general kind

of stochastic integral within the framework of the Bartle in-

tegration theory.





3

The purpose of this dissertation is more modest. We

confine our attention to the case of scalar-valued inte-

grands, anid present a detailed investigation and synthesis

of the appropriate vector measure and integration techniques,

which are then used to exhibit certain kinds of stochastic

integrals as special cases. In carrying out this program we

make improvements in known results concerning the convergence

of sequences of integrals, measures generated by vector

functions, and existence of stochastic integrals.

Chapter I develops the abstract integration theory of

Bartle [1] for the special case at hand, namely, the integra-

tion of scalar functions by finitely additive vector measures.

In this setting more is true than in the general Bartle theory.

For example, Theorem 1.5.8 states roughly that the integral

commutes with any bounded linear operator. This fact enables

us to prove extended forms of the classical convergence

theorems, and sketch a theory of L spaces. We discuss

briefly the case of integration by countably additive vector

measures, and introduce another integral developed by Gould

[16]. This integral is defined by a net of Riemann-Stieltjes-

type sums, as opposed to the Lebesgue approach used in the

development of the Bartle integral by means of simple functions.

We show that these two integrals are equivalent a fact not

discussed by Gould.

Since a stochastic process is a function from an interval

T of the line into an Lp space, as will be discussed in Chap-

ter III, the stochastic integral of a scalar function f on T





4

with res-':ect to z can be thought of as a Stieltjes-type

integral Ff(t)z(dt) in Lp. With this in mind, we inves-

tigate in Chapter II the properties of a Stieltjes measure

m generated by a vector function z on T. We discuss the

boundednzcs of m, and the question of the existence of a

countably additive extension to the Borel sets of T. For

a rather general class of Banach spaces we show in Theorem

11.3.9 that the classical extension theorem for scalar

measures on an algebra is valid. We also give a counter-

example to show that the theorem fails to hold in the Banach

space c Finally we discuss the special case when z takes

its values in a Hilbert space. We consider a theorem of

Cramer [10], which is incorrect as stated, and apply our

results to establish a corrected version. Some implica-

tions of this theorem are then discussed.

In Chapter III we first present some background infor-

mation on probability and stochastic processes, and then

motivate the study of stochastic integration. We show that

the general integration theory of Chapters I and II includes

certain sample path stochastic integrals, the Wiener-Doob

integral, and a martingale stochastic integral. Several

related results are also discussed.













CHAPTER I

Abstract Integration

In this chapter we discuss the integration of scalar

functions by finitely additive vector measures. Section 1

establishes the notation we will use, while Sections 2 and 3

list some basic properties of vector measures. In Section 4

we discuss the space of measurable functions and convergence

in measure. Section 5 is concerned with integration theory

proper, and in Section 6 we apply this theory to prove con-

vergence theorems for integrals. Section 7 outlines part

of the theory of Lp spaces in this setting. Countably addi-

tive measures are discussed briefly in Section 8. Finally,

in Section 9,we compare a Riemann-type integral with the

Lebesgue-type integral of Section 5, and establish the equiva-

lence of these two approaches.


1. Notation. Throughout this dissertation the following

terminology and notation will be used. N is the set of

natural numbers (1,2,3,...) and Nt = (l,2,...,n). The lower-

case letters i,j,k,d, and n will always denote elements of N.

A is a finite subset of N. R denotes the real number field;

we use the usual notation for intervals in R, so that (a,b)

.is an open interval, (a,b] a half-open interval, and so on.





6

X and Y are Banach spaces over the scalar field 0,

which may be either the real or the complex field unless

otherwise specified. I|x denotes the norm of an element

x ( X. X1 is the unit ball of X, that is, X1= [x EX: xl s 11.

B(X,Y) is the Banach space of bounded linear transformations

from X to Y, with the usual topology of uniform convergence.

X* = B(X,f), and XI* is the unit ball of X*. Recall that by

definition,

Ix*l = sup Ix*xl ,
X
x1

while by the Hahn-Banach Theorem,

Ix| = sup jx*xl
X *

Suppose that R and E are families of subsets of a non-

empty set S. R is a ring if R is closed under the formation

of relative complements and finite unions. A ring R is a

a-ring if R is closed under countable unions. E is an

algebra if Z is a ring and S E E. An algebra E is a o-

algebra if E is closed under-countable unions. If 8 is any

family of subsets of S, then a(g) denotes the smallest o-

algebra containing g. If E E R, then n(E,R) denotes the

family of all partitions P = (Ei:1 i Sn} of E whose sets

belong to P. When R is understood we write n(E). If (Ai)

is a sequence of subsets of S which are pairwise disjoint,

we call (Ai) a disjoint sequence. Whenever we write A = EA.

instead of A = LIAi for some set A and sequence (A.), we mean

that (Ai) is a disjoint sequence, and say that A is the sum

of the A.'s. Similarly we write A + B instead of A U B, if

A n B = 0.





7

Suppose that R is a ring and jn: X is a set function.

m is finitely additive if m(A + B) = m(A) + m(B). m is

countably additive if m(FIA.) = Zm(A.) whenever (A.) is a

sequence in R such that EAi E R. We make the convention

that whenever we write m(A) it is assumed that A belongs to

the domain of m. A finitely additive set function defined

on a ring or algebra will be called a measure.

If m:R [o,w] is a set function, then m is monotone if

m(A) : m(B) whenever A c B and A,B E R. m is subadditive if

m(A U B) m(A) + m(B) for every A,B E R. mi 1 m2 means

ml(E) s m2(E) for every E E R.

Standard references for results in measure and integra-

tion theory are Dinculeanu [12], Dunford and Schwartz [15],

and Halmos [17]. We adhere to the notation of Dunford and

Schwartz unless otherwise noted.


2. Properties of Set Functions. In this section we discuss

some elementary properties of measures and their associated

nonnegative set functions. R is a ring and m:R X is a

measure unless otherwise noted. m is bounded if the set

m(R) is a bounded subset of X, that is, sup EE m(E)j < The

total variation v(m,E) of the measure m on a set E E R is de-

fined by
n
v(m,E) = sup E Im(Ei) .
T(E) i=l

It is well known (see Dinculeanu [12]) that the set function

v(m) is finitely additive but may not be finite even though m

is a bounded countably additive measure. As Theorem 2.2 (ii)





P8

infra shows, v(m) is bounded whenever m is a bounded scalar

measure. For this reason v(m) plays an important role in

the theory of integration with respect to a scalar measure m,

since it is a nonnegative measure .'.hich dominates m. When m

is a countably additive vector measure and v(m,E) < c for

each E R, Dinculeanu [12] has shown that most of the results

from the scalar integration theory carry over to the vector

case. In general, however, the total variation may be in-

finite and hence of little value. Thus we must use a more

delicate device known as a control measure for m (see Theorem

3.3 infra). We now introduce a set function i called the

variation of m. For every E 6 R we define

if(E) = sup Im(F) .
F=E
FE6

It is obvious that m is bounded if and only if m is bounded.

The following Lemma will be used several times in this

dissertation.


2.1 Lemma. If aI,...,a are scalars, then

n
E Ia I 4 4 sup I ai .
i=l AGNn iEA
n
In particular, if Xl,...,xn E X and x* C X*, then

n
E Ix*xi. 4 sup I x*x.I S 41x*| sup I xil,
i=l A=Nn iEA A=Nn iA
n
and I a.xi. 4 sup a.il sup I xil.
i=l lsien A=Nn iEC





9

Proof. LetAl, ...,L4 denote the sets of integers i such

that Re a. > 0, Re a. 0, Im a. > 0, and Im a. 0 respec-
1 1 1 1
tively. Then

n n n
S a ail I Re aji + I IIma.i


= E Re a. E Re a. + I Ima.-E Ima.
= Re a. E Re a. + Z Ima. Z Im a
A1 A2 A3 4

=Re E a Re E ai + Im E a. Im E a.
1 A2 A3 A4

4 sup ail
AcN iEA

where we have used the elementary inequalities a I 5 I Re a +

IImal and IRe a limal 5 lal for any a E '. The inequa-

lities concerning the x*x.'s are then immediate. Since
1
n n n
I ai.il = Ix* E a.xil s sup Jai. E Ix*xil
i= 1 i=l igsin i=l

for some x* E XI*, the final inequality follows from the

preceding one. O


As a first application of this lemma we establish the

properties of the variation of a vector measure.


2.2 Theorem. (i) if is nonnegative, monotone, and subadditive.

(ii) If m is scalar-valued, then v(m) s 4fi.

(iii) Im(E) s m(E) s sup v(x*m,E) s 4ff(E), for EE R.
Xl*


Proof. (i) fi is clearly nonnegative and monotone by defini-

tion. If E,F, and G belong to 9 and G Q E U F, then




10

G = (G n E) + (G n (F\E)), so that

|m(G)j = Im(G n E) + m(G n (F\E))

Im(G n E) I + Im(G n (F\E))j

Sfm(E) + mi(F).

Therefore f(E U F) e mi(E) + m(F) for every E,F E R.

(ii) If m is scalar-valued and (E.) E r(E), then

n
E Im(Ei ) 4 sup I m(Ei) | 4 sup Im(F)
i=l A=Nn iEA FeE
FER

by Lemma 2.1. Thus v(m,E) s 4im(E) for every EE R.

(iii) Suppose that E E R. It is obvious that

Im(E) fii (E). Using (ii) we have

fi(E) = sup Im(F)I
FcE
FER

= sup sup Ix*m(F)I
X FcE
FER

sup v(x*m,E)
X1*

4 sup sup Ix*m(F)l
X FCE
FER

= 4i(E). 0


2.3 Remark. By the principle of uniform boundedness (see

Dunford and Schwartz [15]) any function with values in X is

bounded if and only if it is weakly bounded. In particular,

m is bounded if and only if x*m is bounded for every x* E X*.

If R is a c-ring and m is countably additive, then each x*m

is also countably additive. It is known (see Halmos [17])





11

that a countably additive scalar measure on a a-ring is

bounded. Hence m is also bounded. The Orlicz-Pettis Theorem

(see Dunford and Schwartz [15]) states that, conversely, if

x*m is countably additive for each x* E X* and R is a o-ring,

then m is countably additive.

The hereditary ring 3 generated by a ring R is the

family of all subsets of S which are contained in some element

of R. If p:R [o,m] is a monotone set function, then we

can extend p to A as follows. Define

i(E) = inf m (F),
F=E
FER

for every E E M. It is immediate that p is a monotone

extension of p, and p 2 0. If s is subadditive then so is
1
u To see this, suppose that G,H E M and c > 0 is fixed.

Choose E,F E R such that G c E, H z F, and p(E) < p (G) + c,

u(F) < p (H) + e. Then G U H c E U F and so

p (G U H) r p(E U F) p(E) +p(F) < p (G) +p (H) + 2e.
1 1 1
We conclude that L (G U H) p (G) + u (H). In this way we

can extend M to N. A set A E N is said to be an m-null set,

or simply a null set when m is understood, if f(A) = 0. We

shall always identify the extension of ff to by fi unless there

is a possibility of confusion. Note that A is a null set if

and only if for every e > 0, there is an E E 9 such that A c E

and m(E) < c.


3. Strongly Bounded Measures. We now introduce a fundamental

property of certain vector measures, and discuss some of its

implications. R is a ring and m:9 X is a measure unless

otherwise noted.





12

3.1 Definitions. (i) m is said to be strongly bounded

(s-bounded) if for every disjoint sequence (E.) in 2,

lim m(Ei) = 0.


This property was introduced by Ricknar [25], and later

used by Brooks and Jewett [6] to establish convergence

theorems for vector measures. Brooks [4] showed that s-

boundedness is equivalent to the existence of a control

measure. This in turn provides necessary and sufficient

conditions for the existence of an extension of a countably

additive measure from E to o(E) when is an algebra. These

results are presented in Theorems 3.3 and 3.4 infra.


(ii) A series Fx in X is unconditionally convergent

if for every e > 0 there is a L such that A r = 0d implies
E 6
thatI x. < c.
iEti

The following result states some basic properties of an

s-bounded measure. The proof is omitted.


3.2 Theorem. (Rickart [25]) (i) If m is s-bounded then m

is bounded.

(ii) The following statements are equivalent:

(a) m is s-bounded.

(b) For every disjoint sequence (Ei) in R we have

lim m(Ei) = 0.

(c) For every disjoint sequence (E.) in R, Zm(Ei)

is unconditionally convergent.





13

Suppose that r, is a family of measures on R. We say

that the measures in i are uniformly m-continuous, or uni-

formly continuous with respect to m, if for every e > 0

there is a 6 > 0 such that Iu(A) < e for every U in if

Fm(A) < 6. When l contains only one measure p we say that p

is m-continuous. p and m are mutually continuous if each

is continuous with respect to the other.

The most important characterization of s-bounded measures

is the following result due to Brooks [4] which we state with-

out proof.


3.3 Theorem. (Brooks) m:R X is s-bounded if and only if

there is a bounded nonnegative measure m on R such that

(i) m is m-continuous.

(ii) m a m on R.

Moreover, m is countably additive if and only if m is count-

ably additive.


The measure m is called a control measure for m. (i) and

(ii) show that m and m are mutually continuous; it follows

that m and m have the same class of null sets.

A fundamental problem in measure theory is to find con-

ditions under which a countably additive measure on a ring R

has a countably additive extension to the a-ring generated

by R. Using the previous result of Brooks we can prove

an extension theorem for countably additive s-bounded measures

on an algebra. This subject has been investigated by many

authors. Takahashi [28] introduced a kind of boundedness





14

condition less natural than, but equivalent to, the concept

of s-boundedness. His result in the case of an algebra is

therefore included in the following theorem. A measure

m:R X is said to be T-bounded if for every e > 0 there

is an n E N such that if El,...,En are disjoint sets in R,

then Im(Ei)I < E for some i E Nn

To see that T-boundedness is equivalent to s-boundedness,

we remark that if m is not s-bounded then there is a disjoint

sequence (Ei) and an e > 0 such that Im(Ei)I > e, i E N, so

m cannot be T-bounded. This shows that if m is T-bounded

then m is s-bounded. Conversely, suppose that m is s-bounded

with control measure m as in Theorem 3.3. Since m is a

bounded nonnegative measure there is a constant K such that
n
E m(E.) s K for every finite disjoint family (Ei). If m
i=l
is not T-bounded, then there is an e > 0 such that for every

n E N there are disjoint sets Eln,...,Enn in R with m(Ein ,

1 s i s n. This clearly contradicts the boundedness of m, so

m is T-bounded. Since m and m are mutually continuous, m is

also T-bounded.

We now state the extension theorem for measures on an

algebra.


3.4 Theorem. (Brooks [5]) Suppose E is an algebra and

m:E X is countably additive and s-bounded. Then m has a

unique countably additive extension to ao().


Remark. Conversely, if m has a countably additive extension

to a (E), then m is s-bounded on E.





15

Proof. Let m be a control measure for m as in Theorem 3.3.

Then m is countably additive and bounded on E. It is well

known (see Halmos [17]) that a nonnegative, bounded, count-

ably additive measure on an algebra has a unique extension

to a bounded, countably additive measure on a(E) = E1. We

identify this extension by m.

The symmetric difference of two sets A and B, denoted

by A A B, is defined by A A B = (A\B) + (B\A). We also have

A A B = (A U B)\(A n B). The elementary set relations

(i) A C AA B U B A C,

(ii) A U B A C U D A A C U B A D,

(iii) A n B A C n D C A A C U B A D,

(iv) A\B A C\D C A C U B A D,

are straightforward to verify. Define d:E1 x Z1 [O,m) by

d(A,B) = m(A A B). Since m is subadditive, (i) shows that d

satisfies the triangle inequality, and hence is a semi-metric

for E1. Moreover, from the m-continuity of m on E and from

the inequalities m(A\B) s m(A A B) and Im(A) -m(B)I Im(A\B) +

Im(B\A) it follows that m is a uniformly continuous mapping

from E into X.

Since Z1 is the smallest o-algebra containing E, it is

known that for every A E E1 there is a sequence (A ) in Z such

that lim m(A A A) = 0 (see Halmos [17]). This fact shows

that E is a dense subset of El. If we identify sets A and B

in EI whenever d(A,B) = 0, then m is well-defined on E, since

m(A) = m(B) whenever d(A,B) = 0 for A,B E E. Since X is com-

plete, m has a unique extension to a continuous mapping from

E1 into X. We denote this extension by m.




16

To see that m is finitely additive on Z1, we remark

that (ii) (iii), and (iv) show that the mappings (A,B) Ai B,

A n B, and A\B respectively from 771 -E 1 are continuous.

Suppose A,B E 21 and A n B = ,. Let (A ) and (B ) be sequences
1 n n
in Z converging to A and B respectively. (ii) and (iv) imply

that the mapping (A,B) A B is also a continuous function,

so lim d(A B A A B) = 0. By the continuity of m we have

lim m(An A B ) = m(A A B) = m(A + B) since A A B = A + B.

But m(An A Bn) =-m(A\Bn ) + m(B \An), and by the continuity

of (A,B) A\B we have lim m(A \B ) = m(A\B) = m(A) and
n n
lim m(Bn\A ) = m(B). Thus m(A + B) = m(A) + m(B).
n n
Finally, to see that m is countably additive on El it

suffices to show that if Ai.' then lim m(A.) = 0. Since m

is countably additive, lim m(Ai) = 0. This in turn implies

that lim m(Ai) = 0 since m is continuous on the space (Zl,d). O


We quote without proof a final result from the theory

of vector measures. The following theorem is a generalization

of the well-known Vitali-Hahn-Saks Theorem.


3.5 Theorem. (Brooks and Jewett [6]). Suppose R is a o-ring

and v: [0,I ) is a bounded measure. If p :R X is a v-

continuous measure for each n, and if lim un(E) exists for

each E E R, then the pn are uniformly v-continuous.


4. Convergence in Measure and the Space M(m). This section

introduces the concept of convergence in measure for sequences

of functions. Using this concept we can define the class of




17

functions with which the integration theory of Section 5

will be concerned. From now on Z will denote an algebra

and m:Z X is a bounded measure. Then i is a nonnegative,

monotone, subadditive, bounded set function, and, in fact,

these are the only properties that are needed to obtain

the results of this section. F(S) denotes the family of

all scalar functions on S. If f E F(S) we use the abbre-

viation [Ifl > c] for the set (s E S:jf(s)j > ) .


4.1 Definition. A sequence (fk) in F(S) is Cauchy in measure

if for every c > 0 we have

lim [ f.i f. > e] = 0.
i,j-c 1 J

A sequence (fk) in F(S) converges to f E F(S) in measure if

for every e > 0 we have

lim m[I fk fl > ] = 0.
k-c

If (fk) converges to f in measure we write m- lim fk = f.

The following lemma gives a useful equivalent formulation of

convergence in measure.


4.2 Lemma. Suppose f,fk E F(S), k 6 N. Then m- lim fk = f

if and only if there are sets Ak E 2 and a sequence of positive

numbers ek converging to zero such that lim M(Ak) = 0 and

Sfk(s) f(s)I E k if s Ak.

Proof. Let (Ak) and (Ck) satisfy the stated properties, and

suppose e,6 > 0 are given. Choose n such that ek < e and

m(Ak) < 6 if k > n. Since [Ifk fl > e] c [fk fl >Ck] Ak
if k n, we conclude that m[I fk- f > e]< 6 ifk n. Thus m-lim fk=f.




] ?!

Conversely, suppose rn- lim f. = f. For f E F(S),

Dunford and Schwartz [15] define the quantity

lfll = inf (- + fm[i f > e]),
e>0

which is finite since in is bounded. To see that liml[fk fli= 0,

suppose, to the contrary, that liml[fk fl| > 0. Without loss

of generality we may suppose that lifk f|l > c for k E N and

some c > 0. There is an n such that a[I fk f > e/2] < ,/2

if k 2 n. Hence

E < |lfk fll ,/'2 +F[ fk fl > e/2] < e,

which is a contradiction. Therefore limllfk fil = 0. For

-each k E N choose Ek > 0 such that ek + m[Ifk fl > Ek] <

< Ifk fl| + 1/k. Then choose A, F such that [Ifk- fl > k] C Ak

and m(Ak) < m[ fk fl > Ck] + 1/'k. The sequences (Ak) and (ek)

evidently satisfy the properties stated in the lemma. [


If E c S, then IE denotes the indicator function of E,

that is, the function whose value is 1 for s F E and 0 for

s f E. We now define an important class of functions.


4.3 Definition. A E-simple function is any function f E F(S)

which can be represented in the form
n
f = a I ,
i=l i

where a. E 4, and the sets E. E E are disjoint and satisfy
1 1
S = ZE.. When Z is understood, f is called a simple function.

Note that every simple function f has a canonical repre-

sentation with E. = [f = a.].


4.4 Definition. f E F(S) is measurable if there is a sequence

of simple functions converging to f in measure.





19

For f E F(S) and E c S we define the oscillation of

f on E, written 0(f,E), by

0(f,E) = sup If(s) f(t) .
s,t6E

We have the following characterization of measurability.


4.5 Theorem. f is measurable if and only if for every c > 0

there is a partition (E ) E n(S) such that m(E ) < e and

0(f,E.) < e for 2 i s n.


Proof. Suppose (fk) is a sequence of simple functions con-

verging to f in measure. Let (Ak) and (ek) satisfy the

conditions of Lemma 4.2 for f and (fk). Fix e > 0, and

choose k such that i(Ak) < e and Ek < c/3. Suppose that
n
fk = ai IE.; define a partition of S by setting F1 = "k
i=l 1
and F. = Ei_ \Ak for 2 i i s n+ 1. If s,t E F. for some

i > 1, then If(s) f(t) I I f(s) fk(s)I + I fk(t) f(t) l 2c/3.

Hence 0(f,F.) < e for 2 s i s n+l, so (Fi) is the desired

partition.

Conversely, if the conditions of the theorem hold then

choose a sequence of partitions Pk for e = 1/k. If Pk = (Eik)'

set
nk
f = f(sik )I
i=l ik

where sik E Eik for 1 r i s nk. If s X Elk then

If(s) fk(s)| r max 0(f,Ei) 5 1/k. Since mi(Elk) < 1/k ,
2sisnk

the sequences (Elk) and (1/k) satisfy the conditions of

Lemma 4.2, and so m-lim fk = f. We conclude that f is

measurable. ]




20

Let M(m) = M denote the space of all measurable functions

in F(S). Since we have introduced a concept of convergence

in F(S) we can discuss the topological properties of M. The

following result from Dunford and Schwartz [15] lists some

of the useful properties of the space M. We shall omit the

proof.

4.6 Theorem. (Dunford and Schwartz) M is a closed linear

algebra in F(S). If g:' R is continuous, then the mapping

f gof is a continuous function from M into M.

The following result will be very useful in connection with

the Dominated Convergence Theorem.

4.7 Theorem. If (fk) is a sequence of simple functions and

m-lim fk = f, then there is a sequence (gk) of simple functions

such that m-lim gk = f, and Igkl 2|fl on S, for each k E N.

Proof. Let (Ak) and (ek) be sequences as in Lemma 4.2 for f

and (fk). Define gk(s) = fk(s) if s E Ak and Ifk(s)I > 2ek'

and gk(s) = 0 otherwise, for each k. Then if s Ak and

Ifk(s)I > 2Ck, we have gk(s) f() f( < ek. If s Ak and

Ifk(s)l 2ek then
If(s)l If(s) fk(s)| + Ifk(s) I g 3Ek'

so If(s) gk(s)I g 3ek if s A Ak. By Lemma 4.2 we conclude
that m-lim gk = f. If s E Ak or if Ifk(s) s 2ek, then

gk(s) = 0 so certainly Igk(s)I s 21f(s)I. If s e Ak and
SIfk(s)I > 2ek, then







If(s) I fk(s)I fk(s) f(s)

Sfk(s) -

1 /2 fk(s)i = 1/21gk(s) .

Thus Igk(s)I < 21f(s)l for s E S and k E N. ]


5. A Lebesgue-type Integral. We now present the standard

results of integration theory. Many of these theorems were

stated by Bartle [1] in his fundamental paper; they are

included for completeness. Throughout this section Z is

an algebra and m:E X is a bounded measure.

If f is a simple function with representation
n
2 a. IE we define the integral of f over E E 2 by
i=l i
n
SEfdm = a aim(E n E.).
i=l

The following result lists the basic properties of the inte-

gral for simple functions. The set function X(E) = fEfdm is

called the indefinite integral of f.


5.1 Theorem. (Bartle) (i) The integral of a simple function

is independent of the function's representation.

(ii) For a fixed set E E 2 the integral over E is a

linear map from the linear space of simple functions into X.

(iii) For a fixed simple function f the indefinite

integral of f is a measure on E.

(iv) If f is a simple function bounded by a constant K

on a set E, then


|JEfdmJI 4Kmf(E).




22

(v) If U 4 B(X,Y) and f is a simple function, then for

every E CE ,

SEfdUm = U(SEfdm).
Note that by (iv) the indefinite integral of a simple

function is a bounded and m-continuous measure. If m is

s-bounded, then so is the indefinite integral of a simple

function.

The Bartle integral is defined using sequences of simple

functions. The next two theorems discuss the properties of

these sequences which enable us to construct the general

integral. Bartle proved only that (i) implies (ii) in

Theorem 5.2.

5.2 Theorem. Suppose (fk) is a sequence of simple functions

that is Cauchy in measure. Then the following statements

are equivalent:

(i) The indefinite integrals of the fk s are uniformly

m-continuous.

(ii) lim rEfkdm exists uniformly for E E E.


Proof. Suppose (i) holds and e > 0 is given. Choose 6 > 0

such that E E E and ii(E) < 6 implies that for each k E N,

IJEfdml < e.
There is an n E N and sets Ai. E Z such that mf(A ) < and

If.(s) f.(s)I < e/4f(S), if s Aij and i,j > n. Then for

E E and i,j > n, we have

IfEfidm- SEfjdml S ('E\A ij(f- f)dml +

+ ISEnAijf'dml + IErAijf f.dm

s [4e,/4fmi(S)]f(E\\Aij) + 2e < 3e,





23

using (iii) and (iv) of Theorem 5.1. Since E was arbitrary,

we conclude that the sequences (JEfkdm) are Cauchy uni-

formly for E E 2. Since X is complete, (ii) holds.

Conversely, if (ii) holds and e > 0 is given, then

there is an n E N such that for every E E 7,

IE(fk fn)dml < C,
provided k 2 n. Since each indefinite integral is m-continuous,

there is a 6 > 0 such that if A E E and m(A) < 6 then

IJAfkdmI < e,
for 1 s k s n. If fi(A) < 6 and k : n, we also have

SSAfkdm) 1 A (fk fn)dml + (SAfndmI
S2e.

Therefore the indefinite integrals of the fk 's are uniformly

m-continuous. r]

5.3 Theorem. Suppose (fk) and (gk) are sequences of simple

functions such that

(i) m-lim(fk-gk) = 0.

(ii) The indefinite integrals of the fk's and the gk's

are uniformly m-continuous. Then

lim SEfkdm = lim JEgkdm

uniformly for E E S.

Proof. Since hk = fk- gk is a simple function for each k,

and since

Ehkdml < ISEfkdmI + ISEgkdml
for every E E 2, Theorem 5.2 implies that lim r Ehkdm exists

for every E E E. We need only show that this limit is zero






uniformly in E. Fix > 0. There is a 6 > 0 such that if

A E and mf(A) < 6, then for each k G N,

I Zh dm I < e.

Since in-lim h = 0, there is an n E N and sets Ak E E such

that mf(Ak) < 6 and |hk(Z l < e .'4(S) if s Ak, provided

k n. Thus for E E E and k n,

I Ehidml I fE ,khkddmI + IJEgAkdm

S[4c/''i(S) ]m(E\Ak) +

< 2e.

That is, lim Ehkdm = 0 uniformly for E E E. 0


Following Bartle [1] we now define the general integral.


5.4 Definition. f E F(S) is integrable if there is a sequence

(fk) of simple functions such that
(i) m-lim fk = f

(ii) The indefinite integrals of the fk 's are uniformly

m-continuous.


If f is integrable then any sequence (fk) of simple

functions satisfying (i) and (ii) of Definition 5.4 is said

to determine f. Theorem 5.2 shows that if (fk) determines f,

then lim YEfkdm exists uniformly in E. We denote this limit

by the usual symbols

YEfdm or SEf(s)m(ds).
Theorem 5.3 shows that the integral of f is independent of

the determining sequence. Let L(m) denote the family of all

functions that are integrable with respect to m. The following




25

theorem lists some standard properties of the integral. We

omit the proof.

5.5 Theorem. (i) For a fixed set E E E, the integral over

E is a linear mapping from the linear space L(m) into X.

(ii) For a fixed function f E L(m), the indefinite
integral of f is an m-continuous measure on E.

Suppose f E M and there is a null set A such that

supsAI f(s)I < a. Then we say that f is essentially bounded,

and define the essential supremum lfll[ of f by

llfll = inf sup If(s) .
m1 (A)=0 s'A

The following standard result, whose proof refines that of

Bartle, shows that every measurable and essentially bounded

function is integrable.

5.6 Theorem. If f E M is essentially bounded, then f E L(m)

and for every E E E we have

IJEfdml 4l1flim(E)

Proof. Let K = Ilfll, and suppose (fk) is a sequence of simple

functions converging to f in measure. Define

fk(s) Ifk(s) I K,

gk(s) =

(K/i fk(S) )fk(s) I fk(s) > K,
for each s E S and k E N. Then (gk) is a sequence of simple

functions, and

Ifk(s)( K Ifk(s) > K

Igk(s) fk(s) I =
0 {fk(s) I K,




26
since Igk(s) fk(s) I = fk(s)(1-K/Ifk(s) ) = fk (s) K

if Ifk(s)I > K. Suppose e > 0. Since f is essentially
bounded by K, there is a null set A such that If(s)I if s e' A. Now if s ( A and Ifk(s)j > K + e/2, then since

If(s)I : K + e/4, we have

Ifk(s) f(s) I fk(s)1 If(s)

> K + e/2- K- e/4 = c/4.
Therefore [Igk fk > e/2] = [Ifk > K + e/2]

A U [Ifk- fl > e/4].
Finally,

[I9k- fl > e] c [19k- fkl > 3e/4] u [Ifk- fl > /4]
c A U [Ifk fl > e/4].
Since A is null and m-lim fk = f, we conclude that
m-lim gk = f. Since the gk's are uniformly bounded by K,
Theorem 5.1 (iv) implies that for every k,

|fEgkdml < 4Km(E)
for any E E E. This shows that the indefinite integrals
of the gk's are uniformly m-continuous. It follows that
f E L(m), and that for every E E S,

|jEfdm| = limIEfgkdml < 4Km(E). .

Using Theorem 5.6 and the m-continuity of the indefinite
integral of a function f E L(m) we have the following Corollary.

5.7 Corollary. If f E L(m), then the indefinite integral of
f is a bounded measure on E.





27

Proof. Since f is measurable, f is bounded except possibly

on sets of arbitrarily small variation. Choose 6 > 0 such

that if m(E) < 6, then

I(Efdml < 1.
Choose E E E such that m(E) < 6 and f is bounded on S\E, say

by K. For any A E E, therefore

'LAfdml IsA\Efdm + IfAfEfdml

: 4Kf(S) + 1. 0


When E is a o-algebra and m is a countably additive

vector measure, Bartle, Dunford and Schwartz [2] have shown

that the integral satisfies the following property. If

f E L(m) and U E B(X,Y), then f E L(Um) and for each E E,

SEfdUm = U(JEfdm).
We now extend this result to the case where m is a bounded

measure on an algebra E.

5.8 Theorem. Suppose that U E B(X,Y). If f E L(m) then

f E L(Um), and for every E E E we have

SEfdUm = U('Efdm).

Proof. Since IUm(E)I r IU|Im(E) l.it follows at once that

|Um(E)I 5 IUIm(E) for every E E E. From this inequality we
see that if m-lim fk = f, then Um- lim fk = f as well. If

f E L(m) and (fk) is a sequence of simple functions deter-

mining f, then we need only show that the indefinite integrals

of the fk's with respect to Um are uniformly Um-continuous.

Since we have




28

iJ(f ;- fj)dUmI -< IU1SE(fk- fj)dml,
using Theorem 5.1 (v), this follows immediately from

Theorem 5.2 and the fact that lim 3Efkdm exists uniformly

for E E C:. ]


When m is a scalar measure, Dunford and Schwartz [15]

have introduced the following definition of integrability.

f E F(S) is integrable if there is a sequence (fk) of

simple functions converging to f in measure, such that

lim Slfi fjIdv(m) = 0.
1,j- JCO

We shall show that the Bartle concept of integration and that

of Dunford and Schwartz coincide when m is a bounded scalar

measure. We state the following result without proof.


5.9 Theorem. (Dunford and Schwartz [15]) If m is a scalar

measure and f is integrable (in the sense of Dunford and

Schwartz), then the total variation of the indefinite inte-

gral X(E) = Efdm is given by

v(X,E) = SEIf(s)Iv(m,ds).


Using Theorem 5.9 we now prove the equivalence of the

Bartle and the Dunford and Schwartz integration theories in

the case of a scalar measure.


5.10 Theorem. Suppose m is a bounded scalar measure. Then

f E L(m) if and only if f is integrable in the sense of

Dunford and Schwartz. In this case the same sequence of

simple functions determines f for both definitions; conse-

quently the two integrals coincide.




29

Proof. Suppose (fk) is a sequence of simple functions.

By Theorems 5.9 and 2.2 (ii) we have

ISEfidm SEfjdml a ~Efi fjdv(m)

s 4 sup IF(f(i f.)dml
Fq.E
FEZ

for every E E E. Suppose f E L(m) and (fk) is a sequence

determining f. Then by Theorem 5.2 the right-hand term

in the inequality goes to zero uniformly for E C E as

i,j C. Hence

(*) lim SIfi f. dv(m) = 0,
ij-o J'
-and so f is integrable in the sense of Dunford and Schw.artz.

Conversely, if m- lim fk = f and (*) holds, then the left-

hand term of the inequality converges to zero uniformly for

E E E as i,j -. This shows that the sequences ( Efkdm) are

Cauchy uniformly in E, and hence by the completeness of X,

they converge uniformly in E. By Theorem 5.2 we conclude that

the indefinite integrals of the fk's are uniformly m-continuous,

so f E L(m). In either case, the same sequence of simple

functions determines f for both definitions, so the two inte-

grals coincide. [


We now prove some results concerning the domination of

functions by integrable functions.


5.11 Lemma. Suppose (fk) is a sequence of integrable functions,

and g E L(m). If Ifk -. IgI on S for each k, then

SEfkdml 4 sup Ilfgdml
FE
FEE




30

for each E E Z; consequently the indefinite integrals of

the fk's are uniformly m-continuous.


Proof. We use Theorem 5.8 -..ith U = x* E X*, and also

Theorems 5.10, 5.9, and 2.2 (ii) to compute

ISEfkdm = sup SEfkdx*m
X *

Ssup E I fk dv(x*m)
XI *

< sup SElgldv(x*m)
X *

4 sup sup IFgdx*ml
FcE X *
FEZC

= 4 sup fLFgdmI,
FCE
FEZ

for any k E N. Since the indefinite integral of g is m-

continuous, it follows that the indefinite integrals of the

fk s are uniformly m-continuous. 0

5.12 Theorem. If f E M, g E L(m), and Ifi < Igi on S, then

f E L(m).


Proof. Since f E M there is a sequence (fk) of simple functions

converging to f in measure. By Theorem 4.7 we may assume that

Ifkl 21fI on S for every k. Then Ifk < 21gI on S, and
2g E L(m). By Lemma 5.11 we conclude that the indefinite

integrals of the fk's are uniformly m-continuous. Therefore

f E L(m). r


5.13 Corollary. Suppose f c M. Then f E L(m) if and only if

I fl L(m). If (fk) is a sequence of simple functions deter-




31

mining f, then ( fkI) is a sequence of simple functions

determining If .

Proof. The first statement follows from Theorem 5.12. If

fk is a simple function, then so is Ifkl and by Lemma 5.11,

ISEIfkidmI < 4 sup IfFfkdmI.
FEE
FE

Thus the indefinite integrals of the Ifkl's are uniformly

m-continuous, if the indefinite integrals of the fk's are.

Finally, since I fkI I fl I f fl, it follows that

[I Ifk- I fl I > ] c [I fk- f > ] If (fk) determines f,
we conclude that (Ifk ) determines If O

6. Convergence Theorems. In this section we prove stronger

forms of the convergence theorems for sequences of integrals
stated by Bartle [1]. Suppose that f and (fk) are integrable

functions. We say that f and (fk) satisfy property B if

lim SEfkdm = SEfdm

uniformly for E E E.

The main convergence theorem resembles the classical
Vitali convergence theorem.

6.1 Theorem. Suppose f E F(S) and (fk) c L(m). Then f E L(m)
and f, (fk) satisfy property B if and only if

(i) m- lim fk = f.

(ii) The indefinite integrals of the fk's are uniformly
m-continuous.




32

Proof. Suppose f E L(m) and f,(fk) satisfy property B.

To prove that m- lim fk = f we proceed as follows. By

Theorem 5.13, gk = If- fk E L(m) for each k E N. Let

(hkj) be a sequence of simple functions determining gk as

in Definition 5.4. By Theorem 5.13, (ihkjl) is a sequence

of nonnegative simple functions determining Igk = gk, so

without loss of generality we may assume that hkj 2 0 for

each j E N.

By Theorem 5.8 we have gk E L(x*m) for every x* E X*,

hence by Theorem 5.10 gk is integrable in the sense of

Dunford and Schwartz. By Theorems 5.9 and 2.2 (ii) we have

EgkdV' (x*m) = El fk f dv(x*m)

< 4 sup ISF(fk f)dx*ml
FSE
FEZ

S 4 sup I F(fk f)dm ,
FCE
FEE

if x* E X1*. Since f and (fk) satisfy property B, it

follows that given e > 0, there is an n such that if k 2 n

and x* E X1*, then for every E E E,

Egkdv(x*m) < e.
Since (hkj) determines gk, it follows in the same manner

that for a fixed k 2 n, there is an t such that j 2 C implies

that for every E E Z,

SElhkj gkldv(x*m) < ,
provided x* E X *. By omitting the first L-1 of the functions

hkj if necessary, we therefore conclude that for x* E X *

E E E, and k 2 n,







SEh.jdv(x*m) < 2 .
Since hkj is a nonnegative simple function the sets

Ekj = [hkj > y] belong to Z for every y > 0. Moreover,

vv(x*imij) SShkjdv(x*m) < 2e,
so

v(x*m,E.j) < 2e/y,

if x* E X,*. By Theorem 2.2 (iii) we conclude that

m(Ekj) sup v(x*m,Ekj) < 2e/y.
1
Since m-lim hkj = gk, there is an i E N and a set Fk E C

such that fi(Fk) < e/y and Ihki(s) gk(s) 5 y if s Fk.

Therefore, if s % Eki U Fk, we have

Ifk(s) f(s) = gk(s)

Igk(s) hki(s) + Ihki(s)l
S2y ,
and m(Eki U Fk) < 3e/y. Now if 61 and 62 > 0 are given,

then choose y > 0 such that 2y < 61. Then choose e > 0 so

that 3e/y < 62. It follows from the arguments above that

there is an n E N and sets Gk such that if k 2 n, then

fi(Gk) < 62 and Ifk(s) f(s) 61 if s Gk. We conclude
that m-lim f = f.

To see that (ii) holds we note that the argument in the

proof of (ii) implies (i) for Theorem 5.2 applies.

Conversely, if (i) and (ii) hold, then for each k there

is a simple function gk such that T[Igk- fkl > 1/k] < 1/k

and for every E E E,

IEgk fk)dml < 1/k.





34

As in the proof of Theorem 4.6, it follows that m- lim gk= f.

Suppose e > 0. Choose n such that 1/n < e. Then choose

6 > 0 such that i(E) < 6 implies

IJEgkdmI < e and SEfidmi < E
for 1 s k 5 n and i E N. Then if m(E) < 6,

Egkdm I I Egk fkdm + ISE kdm
s 2e
for every k, so the indefinite integrals of the gk's are

uniformly m-continuous. It follows that f E L(m). Since

ISEfkdm- JEfdml I'SE(fk gk)dm + IJ(gk f)dml
S1/k + IjE(gk f)dml,
we conclude that f and (fk) satisfy property B. 0

We now state two important corollaries to Theorem 6.1
which are stronger forms of the results stated by Bartle [1].

6.2 Corollary. (Dominated Convergence) Suppose (fk) c L(m)
and g E L(m). If Ifkl < Ig\ on S for every k E N, then

m-lim fk = f if and only if f E L(m) and f, (fk) satisfy

property B.

Proof. Suppose m-lim fk = f. By Lemma 5.11 the indefinite
integrals of the fk's are uniformly m-continuous, so the
conclusion follows from Theorem 6.1. The converse is immediate

by this same theorem. ]

6.3 Corollary. (Bounded Convergence) Suppose (fk) c M is a
sequence of functions uniformly bounded on S. Then m- lim fk= f
if and only if f E L(m) and f,(fk) satisfy property B.





35

Proof. Since constant functions are integrable, the con-

clusion follows from Corollary 6.2. n


7. Lp Spaces. As an application cf the previous theory, we

present a portion of L space theory. In this section Z is

an algebra and m:E X is a bounded measure.

Recall the following set of inequalities, which were

used in the proof of Lemma 5.11. If f E L(m) and E E E,

then

| EfdmI = sup |jEfdx*mI
X *


< sup SE fl dv (x*m)
X *
1

s 4 sup sup I~Ffdx*ml
FCE X *


= 4 sup IJFfdmi.
FcE


These terms are finite by Corollary 5.7. Let f E M. If

1 ~ p < = and IfP E L(m), define

lfllp = sup [Jf IflPdv(x*m)]l /p
1

Let L denote the space of all functions f E M such that
p
IflP E L(m). Since F c E implies that

ISFfdm|I 5 sup El fldv(x*m),
X *

the inequalities above show that the two expressions

sup [JEI flPdm]1/p l/p
sEup E1 fI Pdm] 1/p and sup [FS IffPdv(x*m)] /p
EEEuI
X *
1!




36
0
are equivalent "norms" for the spaces L We prefer to use
p
the second of these expressions, since the properties of the

scalar norms in L (x*m) are then available.

By Corollary 5.13, it follows that L(m) = LI. Moreover,

since 1 p q < m implies that IflP 1 + Ifl we see

that f E L implies that f E L, by Theorem 5.12. Hence
q p
O LO = LO
L D Lp q
1 p q
From the elementary inequality la+blI 2P(IalI+ IblP),

it follows that If+glP < 2P(IfIp + Ig1P); if f and g are

in L, then so is f + g, by Theorem 5.12. L is clearly
p' p
closed under scalar multiplication, and hence is a linear

space. It is easy to see that llafl = lallfip for any

scalar a. Moreover,

f +gip -= sup[ I f +gPdv(x*m) ]l/p
P X*


< sup[(IfI flPdv(x*m))1/p + (Slg 9v(x*m))l1p]
X *


1 lfllp + llgllp

Since lflp = 0 if f = 0, we conclude that I1l lp is a semi-

norm on L.
P
Suppose 1 < p < and 1/p + 1/q = 1. The HBlder

inequality shows that

IfgI Iflip Igl
S +
Ilf jj4 11gj PllfIIp q jgjjq
fllpllgllq p qlgll

if f E LO and g E L By Theorem 5.12 we conclude that
p q
fg 4 Lo If x* E X1 we have
fg 1L




37

fS fgjdv(x*m) < l\fllpIg. q(1/p + 1/q).

Hence Ilfgll < I flpllgllq
A sequence of functions (fk) in LO is said to converge
k p
in L to a function f E L if we have limlifk flp = 0. The
p p k
following convergence theorem is a consequence of Theorem 6.1.

7.1 Theorem. Suppose 1 s p < m and (fk) is a sequence in L .
Then f E LO and limllfk- flp = 0 if and only if
(i) m- lim fk = f.
(ii) The indefinite integrals of the Ifk P's are uni-
formly m-continuous.

Proof. Suppose f E L0 and limlifk fp = 0. If we set
p p
gk = Ik flp E L(m), then the first part of the proof of
Theorem 6.1 is easily adapted to show that m- lim fk f
Since the indefinite integral of IflI is m-continuous, and

llfkIElp < llfk fllp + lfEllp, (ii) follows as in the proof
of Lemma 5.2.
Conversely, suppose that (i) and (ii) hold. By Theorem
4.6, IfkiP converges to IflP in measure, so by Theorem 6.1,

f E Lp. Moreover, Theorem 4.6 implies that m-limlfk- flP=0.
p
Since Ifk- flp 2P(IfkIp + IflP), we have


EJfk flPdml 5 2p+2 sup iF(IfklP+ IflP)dml,
FcE
FE Z
by Lemma 5.11. Since the indefinite integrals of the ifklP's
are uniformly m-continuous, we conclude that the indefinite
integrals of the Ifk- flP's are also uniformly m-continuous.






By Theorem 6.1, then,
lim JEIfk flpdm = 0
uniformly for E E E. Finally, since

lfk flp < 4 sup fEI fk fl dml,
P EE

we have limllfk fil 0. O

The inequalities at the beginning of this section show
that convergence in L1 is equivalent to property B. For if
f,(fk) belong to Lo = L(m), then

sup IE(fk f)dml < lIfk f1l1 4 supE (fk f)dm .
EE E EE E
Just as Corollary 6.2 followed from Theorem 6.1, we have:

7.2 Corollary. Suppose 1 < p < -, (fk) Lp, g E Lp, and

Ifkl I Ig| on S for each k. Then m-lim fk = f if and only
if f E LO and limllfk f = 0.
p kp

Proof. Since IfklP < Igjl the indefinite integrals of the

Ifk P's are uniformly m-continuous. The conclusions follow
from Theorem 7.1. ]

7.3 Corollary. Suppose 1 s p < c. The subspace of simple
functions is dense in L.
p

Proof. Suppose f E L Since f E M there is a sequence (fk)
of simple functions converging to f in measure, and by Theorem
4.7 we may assume that Ifkl | 21fl on S for each k. By
Corollary 7.2 it follows that limllfk fil = 0. O





39

8. Integration by Countablv Additive Measures. Integration

of scalar functions by a countably additive measure m:Z X,

where S is a a-algebra, has been studied extensively by Bartle,

Dunford and Schwartz [2], and also by Lewis [20]. The Bartle,

Dunford and Schwartz Theory uses a result concerning the

equivalence of weak compactness in the space of scalar

measures with the existence of a control measure (as in

Theorem 3.3), and the Vitali-Hahn-Saks Theorem. By means

of these powerful results the following definition of the

integral yields a theory which coincides with that in Section

5 for countably additive measures.

A sequence of functions (fk) c F(S) is said to converge

(pointwise) almost everywhere to a function f if there is a

null set A such that lim fk(s) = f(s) for every s A. In

this case we write ae lim fk = f.


8.1 Definition. f E F(S) is integrable if there is a sequence

(fk) of simple functions such that ae lim fk = f, and such

that for every E E E, the sequence (fEfkdm) converges in X.

Pointwise almost everywhere convergence replaces con-

vergence in measure in the countably additive situation by

virtue of the nonnegative control measure m. Since m and m

are mutually continuous and countably additive, the standard

theorems on the relation between convergence in measure and

almost everywhere convergence can be shown to hold. In

particular, ae lim f- = f implies m- lim fk = f. Moreover,
K k
since the indefinite integrals of the function fk in Defini-

tion 8.1 are m-continuous meo'ures, the *-quirement that




40

( Efkdm) converges for each E E E, together with the Vitali-
Hahn-Saks Theorem, show that the indefinite integrals of the

fk 's are uniformly m-continuous, as in Definition 5.4.


8.2 Remark. If f is finitely additive, E is a c-algebra, and

m is s-bounded, so that there is a control measure m by

Theorem 3.3, then we may apply Theorem 3.5. Thus if

(fEfkdm) converges for every E, we conclude that the in-
definite integrals of the fk's are uniformly m-continuous.

Convergence in measure must be retained however.


Bartle, Dunford and Schwartz establish a one-sided

form of Theorem 6.1: If (fk) = L(m), ae- lim fk = f, and

the indefinite integrals of the fk's are uniformly m-con-

tinuous, then f E L(m) and lim JEfkdm = 1Efdm for E E 7.

In view of Theorem 6.1 and the Vitali-Hahn-Saks Theorem we

have the following extension:


8.3 Theorem. Suppose (fk) c L(m) and ae -lim fk = f. Then

f E L(m) and f,(fk) satisfy property B if and only if

lim rEfkdm exists for each E E E.


In the case of convergence in L we have by Theorem 7.1:
p

8.4 Theorem. Suppose 1 s p < -, (fk) C Lp, and ae- lim fk f

Then f E L0 and limllfk fl = 0 if and only if lim iE fkiPdm

exists for every E E E.


For future use we state the following useful approxi-

mation theorem for the countably additive case.





41

8.5 Theorem. (Dunford and Schwart.z [15]) Suppose 1 < p < m,

o is an algebra, and E = o(E ). Then the space of -
simple functions is dense in L .
p

Lewis [20] has developed an integration theory for count-

ably additive measures with values in a locally convex space

X using the Pettis approach of employing linear functionals.

This integration theory coincides with that presented above

when X is a Banach space.


9. A Riemann-type Integral. Gould [16] introduced an

integral for scalar functions, which in the case of bounded

functions is defined as a limit of Riemann-Stieltjes-type

sums. Although he calls his integral an improper integral,

it is in fact equivalent to the "proper" Bartle integral

when m is a bounded measure. This alternative approach to

integration has been extended by McShane [22] to obtain a

very general theory. We identify the Gould and the Bartle

integrals as the (G) and the (B) integrals respectively.

In this section E is an algebra and m:E X is a bounded

measure. If P and P' are two partitions in n(S), we write

P' > P to mean that each set in P' is a subset of a set in P;

we say that P' is a refinement of P. If P E n(S) and

P = (E1,...,En), then for a function f E F(S) we introduce

the symbol S(f,P) to indicate any Riemann-Stieltjes-type

sum
n
E f(si)m(Ei),
i=l
where s. E E., 1 s i < n. Let B(S,E) = B(S) denote the family

of all bounded measurable functions on S.





42

9.1 Definition. Suppose f E B(S).. f is integrable (G) if

for any e > 0 there is a P E n(S) such that if P' E n(S)

and P' 2 P, then IS(f,P) S(f,P')j < e.


We omit the proof of the following Theorem.


9.2 Theorem. Suppose f E B(S). f is integrable (G) if and

only if there is an x E X with the following property: For

every c > 0 there is a P E n(S) such that if P' E n(S) and

P' 2 P, then IS(f,P') xj < c.


The vector x is called the integral of f over S. We

now show that for bounded measurable functions the (G). and

(B) integrals coincide.


9.3 Theorem. If f E B(S), then f is integrable (G) and the

(G) and (B) integrals coincide.


Proof. By Theorem 5.6, f E L(m); let x = jSfdm, and let

(fk) be a sequence of simple functions converging to f in
measure. By Theorem 4.7 we may assume that ifkl 21fl on

S for each k. Suppose that If[ s K on S, and fix s > 0.

Choose k E N and A E Z such that (fA) < e/4K and

If(s) fk(s)I < c/4f(S) if s A. Suppose fk = ai I.
Let F = A and F. = E._\A, 2 < i + +1. Then P = (Fi) is

a partition of S and fk is constant over F2,...,Ft+1. If

P' = (B ) > P and if fk(s) = bi for s E Bi, then
n n
IS(fP') SfkdmI = I f(si)m(Bi) JB fkdm
i=1 i=l i
S| E (f(s.) -b.)m(B.) I+I (f(si)-b.)m(B.i)
B. A=,O BicA
4 4[e/4ff(S)]m(S\A) + 4Kff(A) + 8Ki(A)

4c,





43

w.,here w',e have used Lemma 2.1.

Since Ifk < 21fj, Corollary 6.3 shows that f and (fk)

satisfy property B. There is then a k E N such that

|S f3djm xI < e. The argument above shows that there is
a partition P E n(S) such that if P' > P then IS(f,P') -xI 5c.

We conclude that f is integrable (G) and that the (G) and (B)

integrals coincide. 0.


The vector x will now be denoted by ,Sfdm. If E E E

and f Q B(S), then flE is bounded and measurable by Theorem 4.6,

and we write

Efdm = YSf Edm

Suppose f E M(m). The definition of measurability implies

that f is bounded except perhaps on sets of arbitrarily small

variation. Let B(f) denote the family of all sets in Z on

which m is bounded. It is easy to see that B(f) is a ring of

sets. Since f is integrable on each set in B(f), the class

I(f) = (SEfdm:E E B(f)) is nonvoid. Moreover, I(f) is a net

in X over the directed set B(f), where E s F means E C F.

With this motivation we can define the integral for unbounded

measurable functions.


9.4 Definition. f E M is integrable (G) if for every c > 0

there is an E E B(f) such that if F E B(f) and E n F = 0,

then I Ffdmi < e.


In view of the fact that I(f) is a net we have the

following result due to Gould [16].





44

9.5 Theorem. (Gould) f E M is integrable (G) if and only

if there exists an x E X having the following property:

For every e > 0 there is an E E B(f) such that if F E B(f)

and E c F, then ISFfdm xi < c.


Proof. If such an x exists, then since B(f) is a ring and

the indefinite integral of f is a measure on B(f), we have

I Ffdml ISF+Efdm xj + IEfdm xl,

for every E,F e B(f) such that E n F = 0. Fix > 0 and

choose E E B(f) such that if G E B(f) and E C- G, then

IJGfdm xc < ,/2. If F E B(f) and E l F = O, then
E c E+F E B(f), and so

IJFfdml < e.
Conversely, suppose f is integrable (G). Definition 9.4

implies that I(f) is a Cauchy net in X. For suppose e > 0

is given. Choose E E B(f) such that if F E B(f) and E F = 9,

then IfFfdmI < e. Now if F,G E B(f) and F,G B E, we have

IJFfdm fGfdml < IF\Efdml + ISG\Efdm

s 2e,

where we have again used the additivity of the integral. Let

x denote the limit of I(f). It is then immediate that x

satisfies the requirement of the theorem. 0

From this result we see that when f is integrable (G),

then

fSfdm = lim fEfdm
B(f)




45

Suppose now that f E L(m). To see that f is integrable

(G), suppose e > 0, and choose 6 > 0 such that if fi(A) < 6,

then | Afdmj < e. Since f is measurable, there is a set

E E E such that f is bounded on E and m(S\E) < 6. It

follows that for any A E Z such that A n E = d, M(A) < 6,

and so IfAfdmI < e. By Definition 9.4, f is integrable (G).

To see that integrability (G) implies integrability (B)

we need the following result established by Gould [16].


9.6 Theorem. (Gould) If f is integrable (G), then the

indefinite integral of f is a bounded m-continuous measure

-on E.

Proof. Since I(f) is a Cauchy net it is a bounded set, so

the indefinite integral is bounded. Additivity follows from

the additivity of the indefinite integral for bounded func-

tions. To see that the indefinite integral is m-continuous,

suppose c > 0. There is an E E B(f) such that if F E B(f)

and E n F = 0, then ISFfdmI < c. Hence for A E 2, A n E = 0

implies that IfAfdmI | e. On E, however, the indefinite

integral is already m-continuous by Theorem 5.6, so there is

a 6 > 0 such that if M(A) < 6 then IF AEfdml < c. Thus if

m(A) < 6, then

AfdmI I fdm + IfAEfdmI 2e. O

We now prove a result that is the counterpart of

Lemma 5.11.




46

9.7 Lemma. If f,g E M, Ifl < Igl on S, and g is integrable

(G), then for every A E E,

(*) LAfdmIl 4 sup Ifgdm
B=A
BEE

Thus f is integrable (G).

Proof. Suppose E E B(f) and e > 0. The indefinite integral

of f over E is an m-continuous measure by Theorem 5.6, so

there is a 6 > 0 such that if ff(A) < 6, then 'A~EfdmI < e.

Since g is bounded except on sets of arbitrarily small

variation, there is a set F E B(g) such that mf(S\F) < 6.

Since f and g are both bounded on E n F, we have

IJEnFfdml 4 sup Bgadmi

4 sup IBgdml
BcE

by Lemma 5.11. Therefore,

IJEfdm ISE\Ffdm + 'IEnF fdm

C e + 4 sup I[Bgdm
BGE

Since c was arbitrary, we see that (*) holds for every A E B(f),

and hence for every A E E.

To see that (*) implies f is integrable (G), suppose e > 0.

Choose E E B(g) such that if F E B(g) and E n F = 0, then

IfFgdml < e. For any B E E with B n E = 0, we have

IfBgdmiI e. Since B(g) c B(f), we conclude by (*) that if
A E B(f) and A n E = ,, then

IfAfdm| s 4 sup I Fgdml 4e. r
B A
BEZ





47

Suppose now that f is integrable (G). Since f E M,

there is a sequence (fk) of simple functions converging

to f in measure, and by Theorem 4.7 we may assume that

Ifkl < 21fl on S. Since 2f is integrable (G), Lemma 9.7
shows that the indefinite integrals of the fk's are uni-

formly m-continuous, since by Theorem 9.6 the integral of

f is m-continuous. By Definition 5.4, f E L(m).

To see that the (G) and (B) integrals are equal, note

that the definition of the (G) integral as the limit of I(f)

shows that there is an increasing sequence of sets Ek E B(f)

satisfying

(G) Sffdm = lim fSf Ekdm.


Since the functions fTEk are measurable by Theorem 4.6, and

since we can choose the Ek's so that F(S\Ek) < 1/k, it

follows that m-lim fk = f. Since f E L(m) and

If EkI 5 If on S for each k, Corollary 6.2 implies that

(B) Sffdm = lim JSf-Ekdm.


The integrals therefore coincide.













CHAPTER II

Stieltjes Measures and Integrals

In this chapter we specialize the integration theory

of Chapter I to the case where the measure is generated by

a vector-valued function on an interval of the real line.

We shall call such measures Stieltjes measures, and the

resulting integrals will be called Stieltjes integrals.

By restricting the Banach space in which the vector function

takes its values to be an L space over a probability

measure space, we shall study stochastic integration.

(Chapter III). In the present chapter we shall investi-

gate the question of boundedness of a Stieltjes measure,

and the question of the existence of an extension to a

countably additive measure on the Borel sets. Finally we

discuss the important special case of Stieltjes measures

with values in a Hilbert space.


1. Notation. Let T denote a closed, bounded interval in R,

which for convenience we shall assume is the unit interval

[0,1]. Let S denote the family of all subintervals of T

of the form [a,b) or [a,l], together with 0, where we shall

always assume that 0 s a < b < 1. Finally, let E denote

the family of all subsets of T which are of the form A= EAi,

where (Ai) is a finite disjoint family in 9. Note that Z




49

is an algebra of subsets of T, and that the a-algebra

generated by 7 is the family of Borel sets in T, which

is denoted by E1.

Suppose that X is a Banach space, and that z:T X

is a function. Define a set function m:S X by setting

m[a,b) = z(b) z(a) and m[a,l] = z(l) z(a). If A E "

and A = 7Ai, where each A. E 8, then it follows at once

by the definition of m on S that m(A) = Em(Ai). Suppose
n k
now that A E E, and that E Ai and E B. are any two re-
i=l j=l 1
presentations of A by finite sums of sets in o. Then the

sets C.. = A. n B. belong to 8 for each i and j, and we

have

A = E C.., 1 i g n, and B. = T Ci., 1 r j k.
j i
By the preceding remark, it follows that

m(Ai) = E m(C ), and m(B) = E m(Cij),
j J i j
for each i and j. We therefore conclude that

( m(A) = m(C ) = E m(B.).
i i j j

We have just shown that if A E E and A = EAi is any repre-

sentation of A by a finite sum of sets in 8, then we can

define m(A) = E m(A.), and this definition in independent

of the representation of A. It is now immediate that m

is finitely additive on E. In the sequel, we shall associate

with each function z:T X the unique measure m defined as

above on E. When we wish to emphasize that the measure is

generated by z, we write m We call such a measure m the

Stieltjes measure generated by z.





50

If m is a bounded Stieltjes measure on E, then m deter-

mines a class L(m) of functions integrable in the sense of

Chapter I. The Z-simple functions will now be called step

functions; the class of step functions coincides with the

class of all S-simple functions. Every function that is the

uniform limit of a sequence of step functions is obviously

bounded and measurable with respect to any Stieltjes measure

on Z. Hence by Theorem 1.5.6, such functions are integrable,

and the sequence of integrals of the step functions converges

to the integral of the limit function. If m has a countably

additive extension to E1, then Theorem 1.8.5 shows that the

class of step functions is still dense in L(m).


2. Bounded Stieltjes Measures. In this section z:T X is

a function generating the Stieltjes measure m:Z X. We

investigate conditions on z which insure that m is a bounded

measure. By Remark 1.2.3, m is bounded if and only if x*m

is bounded for each x* E X*. Now since

x*mz[a,b) = x*z(b) x*z(a) = mx*z[a,b), for every interval

[a,b) (or [a,l]) in 8, it follows that x*m = m for every

x* E X*. By Theorem 1.2.2 (ii) we know that the scalar

measure m*z is bounded if and only if it has bounded total

variation. Recalling that each set in S is a finite sum of

sets in 8, it follows that
n n
sup E Im(A.) = sup S Im(A) I,
(Ai)cS i=l (A i)C i=l

where each (A.) is a finite disjoint family. That is, the

total variation of any measure m on E is the same as the





51

total variation of m restricted to g. Therefore the measure

m*z is bounded if and only if
n n
sup Z Im z (A )I = sup jx*z(b.) -x*z(ai.) <,
(Ai) i=l (Ai)s1 i=l

where we write each A. as [ai,b.) (or [ai,b.] if b. = 1).

But the right-hand term is just the total variation of the

scalar function x*z. Hence mx*z is bounded if and only if

x*z is a function of bounded variation for each x* E X*.

This establishes the following theorem.


2.1 Theorem. m is a bounded measure on E if and only if the

function z is of weak bounded variation, in the sense that

x*z is a function of bounded variation for x* E X*.


2.2 Remark. Since
n n
sup Im(A) = sup I m(A.) = sup z(b) z(a.) ,
AES (Ai)cg i=l i=l 1 1
where the supremum on the right is taken over all finite

sets of points a. and b., with 0 < a1 b b .. an b n 1,

we have as a corollary to Theorem 2.1 that z is of weak
n
bounded variation if and only if supl S z(b.) -z(ai)I < .
i=1
This result can also be proved directly using Lemma 1.2.1

and the principle of uniform boundedness.

Let C(T) denote the family of all scalar-valued contin-

uous functions on T. If f E C(T), then since f is bounded

and uniformly continuous, it follows from Theorems 1.4.5

and 1.5.6 that f is integrable with respect to any bounded

measure on Z. In fact, if m is a bounded measure and f E C(T),

then in defining the integral according to Definition 1.9.1





52

we may restrict the partitions P to be from n (T,8). To

see this we introduce the concept of the norm of a parti-

tion P E Tr(T,5). If P = (A.:1 < i n), where each A. is
1 1
an interval [ai,b.) (and A = [a ,l]), then the norm of P,

denoted by jPI, is defined as follows: jPI = max (b. -a.).
l~i~n 1 1
We now state the above mentioned result.


2.3 Theorem. If m is a bounded measure and f E C(T), then

for every C > 0, there is a 6 > 0 such that if P and P'

belong to n(T,8) and IP', IPI < 6, then IS(f,P') -S(f,P)| <~.

In particular, this holds if IPI < 6 and P' 2 P.


Proof. By the continuity of f, there is a 6 > 0 such that if

js- t < 6 and s,t E T, then If(s) f(t)j < c/4fi(T). Now

if P,P' E n(T,S) and if PI, IP' < 6/2, then
n k
IS(f,P') S(f,P) = I f(s )m(Ai) S f(t.)m(Bj)I
i=l j=l

n k k n
= | Z f(s.) m(A. n B.) E f(t.) E m(A. nBj)
i=l j=l j=l i=l

n k
= I E (f(si) f(t.))m(A. n Bj) .
i=1 j=l


The only nonzero terms in this double sum occur when

m(Ai n Bj) ) 0; in this case A. n B. 0, and hence

Is. tj. < 6. By Lemma 1.2.1 we conclude that

S(f,P') -S(f,P) s 4[e/4f (T)]m(T) = e.

The final statement of the theorem is immediate. O





53

Motivated by this result and the classical definition

of the Riemann-Stieltjes integral, we make the following

definition. Recall that B(S) is the set of all bounded

measurable functions on S.


2.4 Definition. Suppose m:T X is a measure and f E B(S).

Then f E RS(m) if for every e > 0 there is a 6 > 0 such that

whenever P,P' E n(T,") and Pj, P'j < 6, then

|S(f, P) S(f,P') < C.


Using this definition we could prove as in Theorem 1.9.2

that if f 6 RS(m), then there is a vector x E X to which the

sums S(f,P) converge as |Pl converges to zero. When m is

bounded, Theorem 2.2 shows that C(T) a RS(m), and Theorem

1.9.3 and Definition 1.9.1 show that RS(m) c L(m) and that

the integrals coincide. In fact, if m is any measure from

E to X, and if C(T) c RS(m), then m is bounded. This follows

from Remark 1.2.3 and the following result.


2.5 Theorem. (Dunford [14]) If m is a scalar measure and

C(T) C RS(m), then m is bounded.


Proof. First we remark that if (P ) is any sequence of

partitions such that |P | converges to zero, then for every

f E C(T), the sums S(f,Pn) converge to the integral of f

with respect to m. Suppose that m is not bounded. Since m

is a scalar measure we have only to suppose that the total

variation of m is infinite. It follows that there is a





54

sequence (P ) in n(T,g) such that |PnI < 1.'n and

E Im(Ai)I > n
P
n
for each n, where P = (Ai); we may also assume that m(A) f0.

If for each n we choose a set of points si E A., then by the

preceding remark, lim S(f,P ) = STfdm for every f C(T),

where the sums S(f,P ) are formed using the points (si) chosen

for each P
n
Consider the mapping Un:C(T) X defined by U (f) =S(f,Pn),

n E N. Each U is linear and bounded, and lim U (f) exists
n n
for each f E C(T). Since C(T) with the norm llfl = suPT f(s)|

is a Banach space, it follows by the principle of uniform

boundedness that there is a constant K > 0 such that

(*) IS(f,Pn) I i Kllfll,

for every f E C(T). To obtain a contradiction we shall con-

struct a function f E C(T) such that liflj < 1, but for some
k
n E N, IS(f,P )I > K holds. Choose n such that Im(Ai)( >K,
n j i=l
where P = (A.). Let s., 1 5 i < k, be the sequence chosen

for Pn above. Define f(si) = m(Ai)/Im(Ai) 1 i < k;

f(t) = f(sl) for t in the interval [0,sl); f(t) = f(sk) for

t in the interval [sk,l]; and on each interval [si'si+)

define f to be linear between f(si) and f(si+1). Then

f E C(T), and we have l\fl = 1; however,
k
S(f,Pn) = Z f(s.)m(A.)
i=l
k
= Im(Ai) I > K.
i=l
Since this contradicts (*), m is bounded. ]





55

The following theorem summarizes the results concerning

the boundedness of Stieltjes measures.


2.6 Theorem. Let m:Z X be generated by z. The following

statements are equivalent:

(i) z is of weak bounded variation.

(ii) m is bounded.

(iii) C(T) c RS (m).

(iv) C(T) RS(x*m) for every x* E X*.


Proof.(i) = (ii) follows from Theorem 2.1. Theorem 2.3

shows that (ii) = (iii). We deduce from Theorems 1.5.8 and

2.3 that (iii) a (iv). Suppose that (iv) holds. Theorem 2.5

implies that x*m is bounded; hence, by Remark 1.2.3, m is

bounded. r]


3. Extensions of Countably Additive Measures. If z is a

scalar function on T, then the classical extension theorem

(see Halmos [17]) for the measure m states that m has a
z z
unique countably additive extension to El, the Borel sets

of T, if and only if z is left-continuous and of bounded

(total) variation. When z is a vector function, we shall

show that a similar theorem holds, provided that the Banach

space X does not contain a copy of c We need to introduce

several preliminary concepts and results.

A semi-ring S is a family of subsets of S such that (i)

A,B E c implies that A B E 8; (ii) A,B E S and A B implies

that there is a finite family (C.) in g such that
3-





56

A =C1 c C ... C = B and C..C. 9 for 2 < is n
2 n 1 1-1
(see Halmos [17] or Dinculeanu [12] for information on

semi-rings and measures on semi-rings). If S is a semi-

ring, then the smallest ring R containing ", which h we call

the ring generated by 8, is the family of all sums of finite

disjoint subsets of 8. An example of a semi-ring is just

the class 8 of half-open intervals discussed in Section 1.

In this case the ring generated by 8 is the algebra Z.


3.1 Remark. Suppose that 8 is a semi-ring and that m:8 X

is a set function. It is known (see Dinculeanu [12]) that

if m is finitely additive on 8, then m has a unique finitely

additive extension to R, the ring generated by 8. Moreover,

if m is countably additive on 8, then its extension to R

is countably additive on R as well. From now on, given a

finitely additive set function on 8, we shall assume that m

has been extended to R, and shall denote the extension by m.

When we refer to the variation if we always mean the variation

of the extension, defined as in Chapter I, Section 2, by

rf(A) = sup Im(B) .
BcA
BER

Note that if R is the ring generated by the semi-ring 8,

then we also have
n
ff(A) = sup E m(Bi)I,
(B i)s i=l

where (Bi) denotes a finite disjoint family in 8.




57

If S is a topological space and A s S, then we denote

the interior of A by int A and the closure of A by cl A.

If S is a semi-ring in S and m:8 X is a finitely additive

set function, we make the following definition, with

Remark 3.1 in mind.


3.2 Definition. Suppose that S is a compact space. Then

m is regular if for every E E 8 and c > O, there are sets

A,B E g such that cl A c E c int B and m(B\A) < c.


We now prove a result due to Huneycutt [18] which

generalizes the classical Alexandroff Theorem for regular

scalar measures (see Dunford and Schwartz [15]).


3.3 Theorem. (Huneycutt) If S is a compact space, S is a

semi-ring in S, and m:S X is finitely additive and regular,

then m is countably additive on 8.


Proof. Let (Ei) be a disjoint sequence in S such that

E = ZEi E 8, and fix e > 0. By regularity there are sets

A,B E S with cl A Q E int B and f(B\A) < e. There are

also sequences of sets (A.) and (Bi) in S such that for

each i, cl Ai E. c int B. and f(B.\A.) < e/2'. It follows

that for each n,
n
m[( U Bi)\A] < 2e.
i=l

For
n n
( U B.)\A s [( U B.)\E] U [E\A]
i=l i=1
n
c [ U (Bi\Ei)] U [E\A],
i=l








n n
m[( U Bi)\A] f m[ U (Bi\Ei)] + T[E\A]
i=l i=l 1 1
n
Z E/2 + c < 2e.
i=l


Since cl A is compact and cl A c
k


is a k E N


such that A c U B..
i=l I


n n
Im(A) E m(Ei) < m( U
i=l i=l

n
+ Im( U
i=l


Finally, Im(E)


U Ei = U
i=l i=l


int B., there
1


For n 2 k, therefore,


n
B.) m( U E.) +
S mi=l


Bi) m(A)I
1


n n n
= m[ U Bi\( U E.)]l + Im[( U Bi)\A]
i=l i=l i=l

n n
< M[ U (B.\Ei)] + f[( U B.)\A]
i= 1 i=1

n
S e/2 + 2e < 3e.
i=l
n
- E m(E) j i m(E) m(A)1 +
i=l
n
+ Im(A) S m(Ei)
i=l

S f(E\A) + 3e 4,
w3


if n k. We conclude that m(E) = E m(Ei); hence m is
i=l
countably additive on 8. .


As an immediate corollary of this result we have:


3.4 Corollary.


(Huneycutt [19]) If m:Z X is a finitely


additive measure such that lim m[a -h,a) = 0 for every
h-0
a E (0,1], then m is countably additive on E.





59

Proof. It suffices to show that m is countably additive on

g, and in view of Theorem 3.3 we need only show that m is

regular. Suppose that [a,b) E 9. We may suppose that

0 < a < b < 1, since the cases where a = 0 or b = 1 are

proved in a similar fashion. Choose h > 0 such that

0 < a-h < a < b-h < b. Then we have [a,b-h) E [a,b-h] c

[a,b) r (a-h,b) c [a-h,b), and [ [a-h,b)\,[a,b-h)] !

in[a-h,a) + m[b-h,b). By hypothesis, given e > 0, we can

choose h > 0 such that n~[a-h,a) + m[b-h,b) < e. Therefore

m is regular. r


We now discuss an important class of Banach spaces

characterized by Bessaga and Pelczynski [3]. The signifi-

cance of this class for our purposes lies in the fact that

it is precisely the class of range spaces for vector measures

in which boundedness is equivalent to s-boundedness.

Let co denote the Banach space of all scalar sequences

x = (xn) such that lim xn = with norm defined by

jlx = sunlx nl. If X is a Banach space and (xn) is a

sequence in X, we say_that (x ) is weakly Cauchy if the

scalar sequences (x*x ) are Cauchy sequences for each

x* E X*. We say that (xn) converges weakly to x E X if

lim x*xn = x*x for each x*. Recall that a Banach space is

said to be weakly sequentially complete if every weak

Cauchy sequence converges weakly to some element in X. A

series Exn in X is said to be weakly unordered bounded if

for each x* E X* we have

sup E x*xil < .
A iEA





60

Note that by Lemma 1.2.1, Fxn is weakly unordered bounded

if and only if TIx*xil < c for each x* E X*.

If a Banach space X contains no subspace isometrically

isomorphic to co, then we write X co; otherwise '..e write

X 2 co. We state the following important result without

proof:


3.5 Theorem. (Bessaga and Pelczynski [3]) The following

statements are equivalent:

(i) X co.

(ii) Every weakly unordered bounded series in X con-

verges unconditionally.


The following result was originally stated by Brooks

and Walker [7] when X is weakly sequentially complete, and

slightly extended to the present situation by Diestel [11].


3.6 Corollary. If X A co and m:R X is any measure, then

m is bounded if and only if m is s-bounded.


Proof. By Theorem 1.3.2 an s-bounded measure is always

bounded. Conversely, if m is bounded and (Ei) is a disjoint

sequence in R, then by Lemma 1.2.1 we have
n
Ix*m(E )I < 41x*I sup I m(Ei)I
i=l ANn iEA


< 4 x*[ sup Im(E) < w,
EER

for every n E N and x* E X*. It follows that the series




61

Zm(Ei) is weakly unordered bounded. By Theorem 3.5, there-

fore, Em(E.) converges unconditionally. By Theorem 1.3.2

we conclude that m is s-bounded. ri


3.7 Example. To see that Theorem 3.5 fails to hold when

X C co, consider the sequence of unit vectors en = (6 i) in

c where 6 ni is the Kronecker delta. The series Ze is

weakly unordered bounded, since for any 6 c N, I ei = 1,
iEA
but this series does not converge in c .


We now prove the final result that will be needed for

the main theorem of this section.


3.8 Lemma. Suppose m:Z X is a bounded measure such that

lim m[a-h,a) = 0 for every a E (0,1]. If X c cO, then for
h-0
every a E (0,1] we have

lim m[a-h,a) = 0.
h-0

Proof. Since i is monotone the limit above exists for each

a E (0,1]. Suppose there is an a E (0,1] and a 6 > 0 such

that lim fm[a-h,a) > 6. Let n = 1. In view of Remark 3.1
h-0
there is an n2 > nl and a finite disjoint family of sets
2 1 n2
A2....,A in 8, such that I m(Ai)I > 6. We may assume
2 i=n+l1

for convenience that A. = [a ,b.), where a I i. 1 n n2

If b = a, then choose b such that a < b < b and
"2 "2 "2

Im[b,a)l < 6/2. This is possible by the continuity hypothesis

on m. Then
n2-1 n2
I m(A.) +m[a ,b) I I Z m(A ) m[b,a) > 6/2.
i=n+1 2i=nl+1





62

Redefining b = b, so that A = [a ,b), we have
2 2 2
n2
"2
+ m(A.)| > 6/2.
i=nl+1

Now since M[b ,a) > 6, we can repeat this argument over

the interval [b ,a) to obtain an n3 > n and a finite
n 3 2
disjoint family of sets An ,...,An in S, such that
n3
"3
| E m(Ai) I > 6/2.
i=n2+1

By induction, therefore, we obtain an increasing sequence

(nk) and a disjoint sequence (Ai) in 8 such that for k E N,
nk+l
E m(Ai)I > 6/2.
i=nk+l
nk+1
Define Ek = E Ai; then (Ek) is a disjoint sequence in E,
i=nk+l
and for every k E N,

(*) Jm(Ek)J > 6/2.

Since X 6 co and m is bounded, m is s-bounded by Corollary

3.6. This contradicts (*), so we conclude

lim +[a-h,a) = 0. O
h-40

The following theorem is our main result on the extension

of Stieltjes measures to countably additive Borel measures.


3.9 Theorem. Suppose X 6 c and let E1 be the a-algebra

generated by 2. The measure m :E X has a unique countably

additive extension to Z1 if and only if

(i) z is left continuous on (0,1].

(ii) z is of weak bounded variation.





63

If X 2 c then there is a function z satisfying (i) and

(ii) such that m has no countably additive extension to Ei


Proof. If m has a unique countably additive extension

m' to ZE, then by Remark 1.2.3, m' is bounded, so m is
z z z
also bounded. By Theorem 2.1 z is of weak bounded variation.

Since m is countably additive on 7, lim m (A.) = 0 whenever
Z Z 1
(A ) is a sequence in Z such that Ai' 0. For any sequence

an b where b E (0,1], the sequence [an,b) decreases to 0.

Hence we have 0 = lim m [an,b) = lim [z(b) z(a )], so z is

left continuous on (0,1].

Conversely, if (i) and (ii) hold, then by Theorem 2.1,

m is bounded. (i) implies lim m [a-h,a) = 0 for every
h-0
a E (0,1]. By Lemma 3.8 we have lim+ f[a-h,a) = 0 for
h-0
a E (0,1]. Consequently, by Corollary 3.4, m is countably

additive on Z. Since X c c and m is bounded, m is s-

bounded by Corollary 3.6. We conclude from Theorem 1.3.4

that mz has a unique countably additive extension to Z1.

Now suppose that X 2 co. Let (e ) be the sequence of

unit vectors in co defined in Example 3.7. Let


el n = 1
an=

e Z e. n > 2
i=2

Consider the function z:T c defined by

0 l
z(t) = a I(,/n+l, 1/n] (t)
n= 1





64

for every t E T. One can show that z is left continuous

on (0,1]. Suppose that 0 < s < t < 1. If s = 0, then

since z(0) = 0, we have z(t) z(s) = a provided that

t E (1/n+1,l/n]. If s > 0, then for t E (1/n+1,1/n] and
k
s E (1/n+k+1,l/n+k], z(t) z(s) = an an+k = en+i'
i=1
provided k 2 1. If k = 0 then z(t) z(s) = 0. Now if

[albl),..., [ak,bk) are disjoint intervals in T, with

al < b1 g ... ak < bk, and if a1 > 0, then there is a

finite set nl < n2 ... s n2k of integers such that
n2i
z(b.) -z(a.) = E e., 1 < i < k.
J="2i-1

If al = 0, then z(bl) z(al) = a In either case, since
k n
E [z(bi) z(ai)] is a finite sum of the form E ciei for
i=l i=l
some n, where each ei is 0 or + 1, we conclude that
k
17 (z(b.) z(a ) ] <1.
i=l

By Remark 2.2, z is of weak bounded variation. Now if m
z
has an extension to a countably additive measure on El,

then m is s-bounded. Let A = [1/n+l,1/n) for each n. Then
z n
(A ) is a disjoint sequence in 8, and m (An) =z(1/n) -z(l/n+l) =

en, n E N. Therefore Imz(An) = 1 for every n, so mz is not

s-bounded. Hence m cannot have a countably additive exten-

sion. f]


The requirement that z be left continuous is in a sense

superfluous; if z(t-) exists for every t E (0,1], then we

can define m[a,b) = z(b-) z(a-), (or m[0,b) =m(b-) -m(O)).





65

Essentially this is just normalizing the function z to be

left continuous. Moreover, as the following result shows,

if z(t ) exists at each point of (0,1] and z is of weak

bounded variation, then z determines a unique countably

additive measure on ZE.


3.10 Theorem. If z(t ) exists for every t E (0,1] and if z

is of weak bounded variation, then the function z' defined

by z'(t) = z(t-), t E (0,1], and z'(0) = z(0) is left con-

tinuous on (0,1] and of weak bounded variation.


Proof. Since by definition z' is left continuous on (0,1],

it suffices to show that z' is of weak bounded variation.

If [alb),..., [a ,b ) are disjoint intervals in T with

a1 < b ... s a < b then we can choose points a' < a.
1 1 n n 1 1
(if a = 0 let a' = 0) and b' < bi, such that for each i,
11 1
Iz(ai ) z(a')I < 1/2n and Jz(b ) z(b.') < 1/2n. Then

n n n
S z'(b) -z'(a.)I E z(b.) z(b') + E Iz(ai) z(a.')
i=l 1i=l i=l
n
+ 1 Z z(b') -z(a')
i=l

< 1 + iz (T).
z

By Remark 2.2 it follows that z' is of weak bounded variation.]


If z:T X is a function such that mz has a countably

additive extension to E1, then as we have already seen, z

is left continuous. It follows that right-hand limits also

.exist at each point in [0,1). For suppose a E [0,1) and
S- a Then the intervals a,b) decrease to a) and
bn a Then the intervals [a,bn) decrease to (a] E Z1, and





66

so by the countable additivity of.m mrn(a = lim m [a,b) =
z z z n
lim[z(b ) z(a)]. Thus z(a ) exists. By Theorems 3.10

and 3.9, every function z with right and left-hand limits

that is of weak bounded variation determines a unique

countably additive measure on Zi. Conversely, it is easy

to see that if m is a countably additive measure on -l

with values in X, then there is a function z with right

and left-hand limits that is of weak bounded variation and

satisfies z(a-) = m[0,a) if a E (0,1], and z(a ) = m[0,a]

if a E [0,1). If we normalize this function to be left

continuous and satisfy z(0) = 0, then we have established

a one to one correspondence between the class of all countably

additive measures on E1 and the family of all normalized

functions z that are of weak bounded variation. This is a

generalization of the classical theorem relating countably

additive Borel measures and functions of bounded variation

in the scalar case.


4. Measures in Hilbert Space. In this section we apply

our previous results to sharpen a theorem of Cramer [10],

and discuss its consequences in integration theory. H is

a Hilbert space with inner product (<,*), and z:T H is

function which generates a measure m = m:Z H.
z
:- Recall from Section 1 that the family 8 of all half-

open intervals in T is a semi-ring which generates E. If

8-is any semi-ring, let Sx8 denote the class of all sets

AxB, where A,B E g. It is not difficult to see that 9x8 is





67

also a semi-ring. Elein.ents of gxg are called rectangles.

Consider the set ft:nction ml:gxB 1 defined by

I1 (AxB) = (m(A),m(B)).

By the next lemma, it will follow that mi is finitely addi-

tive on x'g, and therefore by Remark 3.1, mi has a unique

extension to a finitely additive measure on the algebra 3

generated by S8x. 5 is just the family of finite sums of

rectangles in g,..


4.1 Lemma. If A:.-B E Sx and AxB = AlxB1 + A2xB2, then

either A = A = \A2 and B = B. + B2, or B = B = B2 and

A = A1 + A2.


Proof. If A1 = A2, then clearly A = A1 = A2. Suppose that

Bl 0 B2 / ,. If b B1 n B2, then for any a E A,

(a,b) E A XB1 f A2xB2, which is a contradiction. Hence

B1 n B2 = Since B C B1 U B2 c B, E = B1 + B2.

If A1 / A2, then there are al'a2 E A such that

al A \A2 and a2 A2\A Since (a,,bl) and (a2,b2) E AxB

for any bi E Bi, i = 1,2, (al,b2) and (a2',b) E AxB. Since

(al,b2) E A1xB1, B2 c B1. Similarly, B1 z B2, so B= B1 = B2

Then as above we must have A1 n A2 = d and A = A + A2. 5


Now suppose that AxB = A xB1 + A2xB2, and that A= A =A2

and B = B1 + B2. Then

ml(AxB) = m (Ax(B1 + B2))

= (m(A), m(B1 + B2)

= (m(A), m(Bl)) +

= ml(AIxB1) + m (A2XB2)





68

It follows that mi is finitely additive on gxg, and so by

Remark 3.1, we may assume that mI is a scalar measure on J.


4.2 Remark. Suppose that (A.) is a finite disjoint family

in 8. Then we have
n n n
I m(A.) I ( m(A.), Z m(A.))
i=l i=l i=l

n n
= z E
i=l j=l

n n
< 7 E Iml(AixA) .
i=1 j=


We conclude that if mI is a bounded measure on 3, then m is

a bounded measure on Z, or equivalently z is of weak bounded

variation.

Since mI is a scalar measure, Theorem 1.2.2 (ii) implies

that m-l v(ml) 5 4m1 on 3. It follows that in the defini-

tion of regularity, Definition 3.2, we may replace m1 by

v(ml). In what follows we shall use v(ml). The next lemma

is the counterpart of Corollary 3.4.


4.3 Lemma. Suppose mI is bounded. Then mi is regular if

(i) lim v(ml, [s,so)x[t,t)) = 0, so E (0,1], tt0 ET.
s-s
0
(ii) lim v(ml, [s,so)x[t,to)) = 0, t E (0,1], s,s ET.
t-t
o


Proof. Suppose [a,b)x[c,d) E 8x8, and e > 0. By (i) there

is a 6 > 0 such that v(ml,[a- 6,a)x[O,d)) < e and

v(ml,,[b- 6,b)x[O,d)) < e. By (ii) there is an a > 0 such





69

that v(ml, [0,b)x[d -a,d)) < e and v(ml, [0,b)x[c -a,c)) < e.

Now let A = [a,b- 6)x[c,d-a) and B = [a- 6,b)x[c -u,d).

It follows that cl A c [a,b)x[c,d) z int B, and that

B\A c [a- 6,a)x[0,d) U [b- 6,b)x[0,d) U [0,b)x[c -a,c) U

U [0,b)x[d -a,d), so v(ml,B\A) 5 4e. We conclude that mi

is regular, since the same type of argument can be applied

to any set in 8x8. O


We now apply Lemma 4.3 to obtain a connection between

the countable additivity of mi and the continuity properties

of the generating function z.


4.4 Theorem. Suppose that mi is bounded. Then z(t ) and

z(t ) exist for every t E (0,1) (with one-sided limits at

0 and 1). mi is countably additive on J if and only if z

is left continuous.


Proof. Suppose that lir sup jz(t) z(s)I > 6 > 0. Then
Ss ,t-t

there is a sequence s1 < t 1< s2 < t < ... < t such that
1 1 2 2 o
lz(ti) z(si) 2 6, i E N. Since Jz(ti) z(si) 2
2
Im[si,ti) 2 = ml [si,ti)x[si,ti), and since the sequence

([si,ti)x[si,t.)) is disjoint, this contradicts the fact

that m is a bounded scalar measure, and hence has finite

total variation. Therefore we have lim _iz(t) z(s)J = 0.
S : s, t-t
O
Since H is complete there is then a z(to) E H such that
0
lim_ z(t) z(to)j = 0.- The case for limits from the right
t-t

is handled similarly.





70

Suppose that mI is countably additive on U; then m1

is continuous from above at 0. Since [t,to)x[t,t ) decreases

to 5 as t t- we conclude that
0
0 = lim ml[t,to)x[t,to)
t-t
0
o2
= lim z(t) z(t ) 2
t-t
0
Therefore z(to) = z(to)

Conversely, suppose that z is left continuous. In view

of Lemma 4.3 and Theorem 3.3 we need only show

(*) lim_ v(m, [s,so)x[t,to)) =0, so E (0,l],t,to E T,
s-s
0
and similarly for t t as in Lemma 4.3. By Schwartz's in-

equality we have

|ml[s,so)x[t,to) = ((m[s,so), m[t,to))l

S |Iz(so) z(s) z (to) z(t) I ,

and so be the assumption of left continuity of z we see that

(**) lim_ ml[s,so)x[t,t ) = 0
s-s
0
for s E (0,1] and t,to E T. Since v(ml) is monotone, the

limit (*) exists for every s E (0,1] and t,t E T. Suppose

that for some choice of so,t, and t we have

lim_ v(ml,[s,s )x[t,tO)) > 6,
s-s
0o
for some 6 > 0. Then there is a family of disjoint rectangles

[ai,b.)x[ci,d.), 2 i n2, contained in [0,s )x[t,to), such

that
n2
"2
SIml[ai,bi)x[c ,di) > 6.
i=2





71

We shall now show that we may assume b. < s for each i.

Suppose that [a,b)x[c,d) is any rectangle in [O,so)x[t,to)

such that b = so, and

Iml[a,b)x[c,d)| = a > c > 0,

for some c > 0. By the continuity condition (**) we can

select a b' such that a < b' < s and
0
Iml[b', so)x[c,d) < a- c.
Then Iml[a,b')x[c,d) I > Iml[a,b)x[c,d) Iml[b',so)x[c,d)

> a (a c) = c.

Now if any of the b. = s then the argument above shows

that we can find new b.'s such that b. < s and
1 1 O
n2
"2
E Iml[ai,b.)x[ci,di.) > 6.
i=2

Let 6 = min (s b.); then v(m, [s 61 s )x[t,t )) > 6.
2i 0 1 o 0 o o

so we can repeat the process above. By induction we obtain

a disjoint sequence (A xBi) of rectangles and an increasing

sequence (nk) of integers (set nl = 1) such that
nk+1
S ml(AixBi) > 6,
i-=nk+l
contradicting the boundedness of mi. We conclude that (*)

holds. The proof for t-t- is similar to the above. O
0

The theorem we are about to prove was essentially

stated by Cramer [10], but his statement and proof contain

a gap which we now rectify. In our terminology, the statement

of Cramer's result is as follows. Let z:T H be a function

generating m and mi as above. If mi is a bounded measure,





72

then m and mi have countably additive extensions to Z1 and

.1 respectively, where U1 denotes the a-algebra generated

by U. Since even for scalar functions the measure m is not
z
countably additive unless z is left continuous, the theorem

as stated is not correct. We now present the corrected

version.


4.5 Theorem. If mI is bounded, then the measures m and m1

have unique countably additive extensions to Z1 and 51

respectively if and only if z is left continuous. In this

case we have

(*) (m(A), m(B)) = m (AxB)

for every A and B in El, where we identify the extensions by

m and mi.


Proof. If mi has a countably additive extension then z is

left continuous by Theorem 4.4. Conversely, if z is left

continuous, then by this same theorem mi is countably addi-

tive on J. By the assumption of boundedness, mi has a unique

extension to 31 (by the classical extension theorem or Theorem

1.3.4), and z is of weak bounded variation by Remark 4.2. By

Theorem 3.9, m has a unique countably additive extension to

1'
To see that (*) holds, fix B E E, and define set functions

p and X on E1 as follows:

p(A) = ml(AxB), X(A) = (m(A), m(B)),

for A E Z1. Since m and mi are countably additive, it follows

that p and X are finitely additive and continuous from above

at 0; thus p and X are countably additive scalar measures.




73

Since X(A) = u(A) for A E Z, it follows by the uniqueness of

extensions that X(A) = p(A) for A E ,El or m (AxB)= (m(A),m(B))

for A E and B E E fixed and B E E1 fixed. Repeating this

argument for A E 71 fixed and B E gives the desired result. M


The author is indebted to Professor Brooks for dis-

cussions on the previous material; in particular the proof

of (*) in Theorem 4.5 is due to him.

As an application of Theorem 4.5, we discuss the most

important case, namely functions z with orthogonal increments.

A function z:T H has orthogonal increments if

(z(tI) z(sl), z(t2) z(s2)) = 0 whenever s < tl < s2 < t2.

The differences z(t) z(s) are called the increments of the

function z. This property simply states that for disjoint

sets A,B E 8 we have m (AXB) = (m(A), m(B)) = 0; mI therefore

reduces to a nonnegative measure on S if we define

ml(A) = (m(A), m(A)) = im(A) 2. It follows from orthogonality

that mI is finitely additive, since

ml(A + B) = (m(A+B), m(A+B))

= (m(A), m(A)) +(m(A), m(B)) +

+ (m(B), m(A))+ (m(B), m(B))

= m1(A) + m (B).

Since mI is nonnegative and finitely additive it is monotone,

and so m (A) 5 m (T) for every A E E. By Theorem 4.5 we

conclude that m and mi have unique extensions to El if and

only if z is left continuous. Moreover, since mI is bounded,

Theorem 4.4 implies that z(t+) and z(t-) (z(0 ) and z(l-))

always exist. It follows that any function z with orthogonal





74

increments determines countably additive measures m and ml

if we normalize z to be left continuous. In fact, z is

continuous except possibly at countably many points, since

F(t) = ml[O,t) is a nondecreasing function and Iz(t) -z(s)12=

F(t) F(s).

Suppose now that z is any function with orthogonal in-

crements, not necessarily left continuous, and z generates

the measures m:E H and m:E [0,-) as above. Since

[m(A)]2 = sup Jm(B) 2 = sup ml(B) = m (A),
BcA BQA
BEE BEE

it follows that convergence in measure with respect to m is

equivalent to convergence in measure with respect to mi.

Moreover, if (fk) is a sequence of step functions on T, then

the indefinite integrals of the fk's are uniformly m-continuous

if and only if they are uniformly ml-continuous. Suppose
n
f = E a.IE is a step function; then
i=l E.
n n
I Efamd2 = (< a.m(E n Ei), Z a.m(E n Ei))
i=l i=l

n 1 -
= ai 2ml(E n Ei) = Elfl2dm
i=l

using the orthogonality of the measure m. By the remarks

above, and by Definition 1.5.4 and Theorem 1.7.1, we conclude

that f E L(m) if and only if f E Lo(ml). Moreover, for

step functions f and g it can be shown that

(SEfdm, IFgdm) = EnFfgdml'

and so by continuity this equality holds for every f and g in




75

L(m) = Lo (m1). If z is left continuous, so that m and m

have countably additive extensions to Borel measures, then

by Theorem 1.8.5 the class of step functions is dense in

L(m) = L (m) and the results above still hold by con-

tinuity.















CHAPTER III

Stochastic Integration

In this chapter we shall restrict the integration

theory of the preceding chapters to the case where the

Banach space X is an L space over a probability measure

space. Stieltjes integrals with respect to a function

z:T Lp are stochastic integrals in the sense that the

integral is an element of L or in probabilistic terms,

a random variable. We show that the general integration

theory of Chapters I and II includes certain stochastic

integrals previously defined by other methods.

In Section 1 we introduce and discuss the probabilistic

concepts which are used. Section 2 consists mainly of back-

ground material on the theory of stochastic processes. In

Section 3 we attempt to motivate the study of stochastic

integration, and then discuss the general integration theory

in the present context. The sample path stochastic integral

is studied in Section 4; we compute some statistical pro-

perties of the integral in a simple special case, and discuss

an integration by parts formula. The Wiener-Doob stochastic

integral in L2 is discussed briefly in Section 5. In Section

6 we deal with martingale stochastic integrals, and prove a

general existence theorem.





77

1. Probability Conceots and notation. We say that (1,F)

is a measurable space if 0 is a nonvoid set (w is a generic

element in in D) and F is a a-algebra of subsets of n. If

there is a countably additive measure P defined on F with

values in [0,1], and if P(n) = 1, then we say that (G,F,P)

is a probability measure space, and P is a probability

measure. A probability measure space, or probability space,

is therefore just a finite nonnegative measure space with

the property that 0 has measure 1.

Suppose that (Q,F) and (Q',F') are two measurable spaces,

and x:O n' is a function. x is said to be measurable

(relative to F and F') if the set [x E A'] x- (A') belongs

to F for every A' E F The family of all sets [x E A'],

A' E F', is a a-algebra contained in F, and we denote it by

F(x). Note that F(x) is the smallest a-algebra F" contained

in F such that x is measurable relative to F" and F'. If P

is a measure on F, then a measurable function x determines a

measure P' on F' by the relationship P'(A') = P[x E A'] for

each A' E F'. If P is a probability measure on F then P'

is a probability measure on F' as well. In this case P' is

called the distribution of x. When n' = 1 we shall always

take F' to be the a-algebra of Borel sets in 0.

A measurable scalar-valued function on a probability

space (0,F,P) is called a random variable. As motivation

for this term, suppose we have a system whose state can be

described by a scalar variable x, and whose behavior is

subject to some sort of statistical variation. We imagine





78

the numbers x corresponding to the (random) states of the

system to be the values of a function x defined on some

probability space. The assumption that x is a measurable

function insures that the probability of each event x E B

is defined, where B is any Borel set. This probability is

just P[x E B].

Lp L (Q,F,P), where 1 s p < -, will denote the Banach

space of all equivalence classes of random variables x such

that

E xlPx n Mx(U) IP(d,) < m.
L, denotes the space of equivalence classes of essentially

bounded random variables. It is known (see Dunford and

Schwartz [15]) that for 1 p < -, L is weakly sequentially

complete. Since co is not weakly sequentially complete, it

follows that Lp j co; hence Theorem 11.3.9 applies to these

spaces. The number ElxlI is called the p-th absolute moment

of x. If k E N, then Ex is called the k-th moment of x.

In particular, the first moment Ex is called the mean or

expectation of x. It is well known from the theory of in-

tegration that the expectation operator E is a continuous

linear functional on L1. If x E L2, then the number
2 2#
(2(x) = EIx-Ex2 = ElIx2 Ex2

is called the variance of x, and represents in some sense the

dispersion of x about its mean value Ex.

Standard references for results in probability theory are

Love [21] and Chung [9]; most of the material in this first

section can be found in either of these works.





79

2. Stochastic Processes. In this section we discuss the

definition of a stochastic process (random function) as a

mapping from a subset T of R into an L space over a pro-

bability space. In order to illustrate some of the important

properties that a process may have, we discuss two classical

examples, the Poisson process and the Wiener process.

Let T be a subset of R whose elements are interpreted

as points in time, and suppose we have a system whose

observable state at time t E T is some scalar quantity x(t).

If the system is subject to influences of a random or

statistical nature, then we are led to suppose that x(t) is

a random variable for each t E T. Our mathematical model

of the system is then a family of random variables x(t), t E T,

which are defined on some underlying probability space (0,F,P).
/
If we assume that each x(t) belongs to Lp, so that the con-

vergence properties of this space are available, then we have

defined a mapping x from T into L x is called a stochastic

process or random function since it represents the behavior

of a system subject to stochastic (random) influences.

Since Lp is a space of (equivalence classes of) functions

from Q into , we may also think of a random function x as a

mapping from TxO . Then each w E n determines a function

x(*,w):T 4; this function is called the sample function or

sample path corresponding to w. Every set A E F determines

a set of sample functions (x(.,wu):w E A), and we say that the

probability of this set of sample functions is p if P(A) = p.





80

A fundamental problem in the theory of random functions con-

cerns the fact that we would like to discuss the probabilities

of sets of sample functions which may not correspond to sets

in F as above. For example, given any finite set of times

tl'...tn and Borel sets B1,...,B n the set
n
[x(t.) E B.:l i=l 1

is in F. Even for countably many t.'s and B.'s such sets are
1 1
in F. However, the set

[x(t) E B:t E (a,b)] = n [x(t) E B]
tE(a,b)

clearly need not be measurable. Doob [13] discovered a

reasonable way to skirt this issue. He was able to show that

every random function x is equivalent to a random function y

which behaves nicely in this regard, where equivalence means

that for every t E T, P[x(t) = y(t)] = 1. We shall always

assume for the sake of simplicity that we are dealing with

such well-behaved processes.

In order to discuss two important examples of stochastic

processes we need to introduce a few additional concepts.

Suppose that x(t), t E T,is a random function, and tl,...,tn

is a finite subset of T. Assume for simplicity that x(t) is

real-valued for each t. Let Rn denote real Euclidean n-space,

and-let Bn denote the a-algebra of Borel sets in Rn. It can

be shown that the family of all product sets Bx...xB n, where

B. C B for 1 s i < n, is a semi-ring which generates B.

Consider the mapping w [x(tlW),...,x(tn,)],-which is a





81

function from 0 into Rn. Since each x(ti) is measurable,

it follows that the set
n
[[x(t ),...,x(t )] 6EB >.... ,xB ] = [x(ti) E Bi]
i=1

belongs to F, for every product set Blx...xB n As in Section

1, the mapping w [x(tlw),...,x(tnw)] determines a measure

on the semi-ring of product sets, and by a standard extension

theorem (see Halmos [17]), this measure extends to a probability

measure P on Bn. The measure P is called the
tmeasut tl..t1 .t
tl'''tn 1'*,'' n
joint distribution of the random variables x(tl),...,x(t ).

It seems reasonable, and in fact can be verified (see Loeve

[21]), that a random function is characterized by the family

of all its joint distributions Ptl

Suppose that x(t), t E T, is a random function. As in

Section 11.4, we say that x(t2) x(tl) is an increment of the

process. x(t) is said to have stationary increments if for

every choice of t > 0 and s < t s < t2 ... sn < t in
1 1 2 2 n n
T, the joint distribution of the increments x(ti + t) -x(s + t),

1 s i s n, coincides with the joint distribution of the incre-

ments x(ti) x(si), 1 s i < n. In other words, the prob-

abilistic properties of the increments are invariant under a

shift in time.

Recall that every random variable x(t) x(s) determines

a a-algebra F(x(t) x(s)) as in Section 1. We say that

x(t), t E T, has independent increments if for each choice

of sI < t 1 ... sn < tn in T and Ai F(x(ti) -x(si)),
n
1 s i n, P[ n Ai] = P(AI) ... P(A ). In this case it is
i=l





82
n
known (see Loeve [21]) that if E TT x(ti) x(si) < m,
i=l
then we have
n n
E T [x(t.) x(s )] = TT E[x(t.) x(si)].
i=l i=l

Now we consider a classical random function called the

Poisson process. Suppose we have a general situation where

events occur in a random fashion that is in some sense uni-

form over T = [0,1]. Let x(t) denote the number of events

which have occurred in the interval [O,t). It seems reason-

able to assume that the random function x has the following

properties:

(i) Independent increments. The increment x(t) -x(s),

t > s, represents the number of events occurring in [s,t),

and we would expect that over disjoint intervals of time,

the numbers of events occurring in those intervals are in-

dependent of each other.

(ii) Stationary increments. The uniformity assumption

means that over disjoint intervals of the same length, we

expect that the distribution of occurrences is the same.

(iii) x(0) = 0.

If in addition we assume that there is a X > 0 such that for

small values of h > 0 we have

P[x(h) = 1] = Xh + o(h),

and

P[x(h) = 0] = 1 Xh + o(h),

then


P[x(h) > 1] = o(h),





83

and it can be shown (see Doob [13]) that for every s < t

in T, we have
,(t-s) n (t s)n
P[x(t) -x(s) =n] = e-(- n(-s)
n:

for n = 0,1,2,... This discrete probability distribution

is called the Poisson distribution, and the relation above

states that the random variable x(t) x(s) is distributed

according to a Poisson distribution, with parameter .. The

mean and variance of the increments are given by

E[x(t) x(s)] = X(t-s),

and

2 (x(t) -x(s))= E[x(t) -x(s)] [E(x(t) -x s))]2

= (t s).


The sample functions of a Poisson random function

(except for a set of sample functions of measure zero) are

monotone, nondecreasing, nonnegative, integer-valued functions

of t, with at most finitely many jumps of unit magnitude in

T. This might be expected in view of the event-counting

interpretation of x(t).

A second important example of a random function is the

Wiener process, which has been used extensively as a mathe-

matical model for Brownian motion. Suppose we observe a

particle undergoing Brownian motion, and restrict our atten-

tion to, say, the x coordinate of its position. Due to the

random bombardment of the particle by the fluid's molecules,

this x coordinate changes in an erratic fashion as time

progresses. If we denote the value of x at time t > 0 by





84

x(t), then it seems reasonable to assume that x(t) is a

random function which has the following properties:

(i) Independent increments. The increment x(t) -x(s)

represents a change in position along the x-axis over the

interval [s,t). Over disjoint intervals of time these

changes should be independent of each other due to the ran-

dom character of the molecular bombardments.

(ii) Stationary increments. Since we would assume that

the fluid in which the particle is suspended is homogeneous,

the distributions of change over two intervals of the same

length should be the same.

(iii) x(0) 0, assuming we use the position of our

initial observation as the center of the coordinate system.

(iv) Normally (Gaussian) distributed increments, with

mean zero and variance proportional to the length of the

time interval. This follows from the assumption that the

motion is random and continuous, and means that if a < b,

then

P[a < x(t) -x(s) /27c2 (t-s) a

where s < t, and o > 0 is some constant, namely the variance

over an interval of unit length (we have assumed t 0

so T = [0,-))

It can be shown that except for a set of sample functions

of measure 0, the sample functions of a Wiener process are

continuous, nowhere differentiable, and of infinite variation

over any finite interval. Although this last property runs





85

counter to what intuition says the sample paths should look

like, the Wiener process has played a very significant role

in the application of probability theory to physics and

engineering. In fact, the first stochastic integrals were

defined by Wiener for scalar functions with respect to a

Wiener process.

A standard reference for results on stochastic processes

is Doob [13]. Much of the material in this section can be

found in his classic work.


3. Stochastic Integrals. We now discuss as example which

motivates the theory of stochastic integration. Suppose we

have a system consisting of a signal processing unit with

an input signal z(t) and an output signal x(t). Suppose

that the effect of the processor can be described by a

scalar function of two variables g(t,T) in the following

way. A small change z(dT) = z(T + dr) z(T) in the input

at time T produces a change in the output of the system at

a time t 2 7 given by g(t,T)z(dr). Moreover, suppose that

the response of the system is linear, in the sense that

small disturbances z(d 1),...,z(diTn produce a change in
n
the output given by E g(t,T.)z(dT.), for t 2 max T.. If
1 1 1
i=l
we assume that g(t,T) = 0 if t < T, then for a given input

function z(t), we have formally that

x(t) = Jf g(t,T)z(dT).

As long as the functions g and z are sufficiently well-

behaved scalar functions, the definition of this integral

poses no problem.





86

Suppose, however, that the input z(t) is a random

function, for example, a signal contaminated by random

noise. Then the output will also be random, and we are

faced with the problem of how to define the integral above

so that it is a random variable for each t. If the function

g is continuous in T for every t, and if almost every sample

function of z is a function of bounded variation, then we

could define the random output in a pointwise fashion as

fo 1 low0s:

(*) x(t,w) = SS_ g(t,T)z(dT,u).

Stochastic integrals defined in this way are called sample

path integrals; although they are the most restrictive kind

of integral, their study does provide motivation for the more

powerful kinds of stochastic integrals. In general, this

simple solution to the problem is not feasible, since even

processes as well-behaved as the Wiener process have almost

every sample function of unbounded variation.

Using the integration theory of Chapters I and II when

the Banach space X is an Lp space over a probability space,

we obtain a very general definition of the stochastic inte-

gral. Suppose that z:T Lp is a random function, where

1 < p < m. As in Chapter II we can define a measure m:7 Lp

by setting m[a,b) = z(b) z(a) on S. For each A E E, m(A)

is thus a random variable in Lp. The measure m is bounded

if and only if z is of weak bounded variation, and by the

form of Lp* this is equivalent, using Theorem 11.2.1, to





87

requiring that the scalar functions t E[z(t)x] are of

bounded variation for every x E L where 1/p + 1/q = 1

if 1 < p < -, and q = if p = 1.

Suppose that z is of weak bounded variation. Then z

determines a class L(z) L(m) of integrable scalar functions,

and for f E L(z) we write

Sf(t)z(dt) = STf(t)m(dt).
We shall call such integrals stochastic Stieltjes integrals,

or norm integrals for short, since they are defined as the

limit of a sequence of integrals of step functions in the L

norm. If |Xllp denotes the L norm (ElxlP) /p of an element

x E Lp, then it is known (see Loeve [21]) that 1 r s p and

x E L implies that llxllr s lXll This shows that Lr 7 L ,

and that the canonical embedding Up of Lp into L is continuous
pr p r
By Theorem 1.5.8 it follows that if f E L(z) then f E L(Uprz)
and

Upr(SEfdz) = SEfdUprz
That is, if f is integrable with respect to z as a mapping into

Lp for 1 p < -, then f is integrable with respect to z as a

mapping into Lr, 1 r : p, and the integrals coincide. More-

over, it is clear that the same sequence of step functions

determines the integrals of f in Lp and in Lr.

By Theorem 1.5.8 we have

E(jTf(t)z(dt))x = STf(t)E[z(dt)x],
for every x E Lq, since as is well known, this is the form

of linear functionals on Lp (see Dunford and Schwartz [15]).







Tr.us

E(STf(t)(d)) = Tf(t)E[z(dt)]
Also

E Tf(t)z(dt) 2 = E[Tf(t) z(dt) ] [Tf(T)z(d ) ]

= Tf(t)E[z(dt)JTf(T)z(dT) ].
In general

E[fjf(t)z(t))]k = Tf(t)E[z(dt) (STf(T)z(dT))k-1,

since [ Tf(T )z(dT)]k-1 belongs to Lk/k_- if z:T Lk. If

we evaluate the scalar functions

t -E[z(t) (STf(T)z(dT))k-] ,

then we can compute the k-th moment of the integral by per-

forming a scalar integration. In practice this may be of some

value.

4. Sample Path Integrals. Let T = [0,1] and g be a continuous

function on T. Let z(t). be a Poisson process on T with para-

meter .. As described in Section 2, almost every sample path

z(.,w) is a monotone nondecreasing function, with z(0,w) = 0,

and finitely many jumps of unit magnitude. We can therefore

define the Riemann-Stieltjes integral

S. g(t)z(dt,uw)
for almost every w. By Theorem 11.2.3 it follows that

1 g(t)z(dt,w) = lim Z g(i-1/2n) [z(i/2n, ) z(i-1/2n u1)]
n-- i=l

for almost every w. Since the summands g(i-1/2n) [z(i/2) -

z(i--/2n)] are random variables it follows that the sample





89

path integral 1 g(t)z(dt) is also a random variable.

Moreover, we can evaluate the integral directly, due to the

simple behavior of the sample functions. Fix w and let

n = z(l,u1). Let tl,...,t denote the n points of jump of
o 1 n o
o
the sample function z( ,w). For every n and i, the difference

z(i/2n,w) z(i-l/2,uw) is either 1 or O, depending on whether

one of the t.'s belongs to (i-1/2n,i/2n] or not. By the con-

tinuity of g we see that
n
o
Si g(t)z(dt,w) = E g(ti).
i=l

In general, denote by ti () the point in T of the i-th jump

of the sample function z(*.,) ; that is, for t s ti.(),

z(t,w) i -1, while for t > ti(w), z(t,w) 2 i. Since

[w:ti (u) < a] = [z(a) 2 i] E F for each a E T, t. is a random

variable for each i. Thus

1 z(l,w)
JO g(t)z(dt,w) = g(t (w)).
i=l

That is, the integral of g with respect to the Poisson process

z is the sum of a random number (z(l,w)) of random variables

of the form g(ti(w)).

S We can formally compute the mean of the integral as

follows: we have
2n 2n
E[ E g(i-l/2) [z(i/2) /2-Z(i1,n)]= g(i-1/2n) (1/2n
i=1 i=l
2n
Since lim E g(i-1/2"n)(1,/2) = 1g(t)dt, and since the
i=l
2n
functions E g(i-1/2n )[i-,,2nI/2n) converge uniformly to
i=l [i-/2nl/2
g on T, we would expect that

E[j g(t)z(dt)] = X1 g(t)dt.





90

This will follow once it is established that the stochastic

Stieltjes integral exists. Proceeding to the formal com-

putation of the variance of the integral, we compute
2n
El Z g(i-1/2n) z(i/2n) z(i-1/2)]2 =
i=l

2n 2n
= E E g(i-1/2n)g(j-l/2n)E[z(i/2n) -z(i-l/2n) i x
i=l j=l


x [z(j/2n)-z (j-l/2n)

n
2
= g(i-1/2n) 2 E[z(i/2n)-z(i-l/2 )] +
i=l
nn
+ E g(i-1/2n)g(j-l/2n) E[z(i/2n)-z(i-1/2n)] i
ifj=1


x E[z(j/2n) z(j-1/2n)],

since z has independent increments. To evaluate the second

moments, we suppose that x is a Poisson random variable with

parameter a. Then

- n C n-i
2 2 -a a -a a
Ex = n = ae L n n-
Sn=0 n= 1

C- n-1 m n-1
= ae-a[ (n-1) (n-l) + E (n-l),


-a a a
= ae [ae + e ] = a(a + 1).

Since z(i/2n) z(i-1/2n) is a Poisson random variable with

parameter \2-n, we conclude

E[z(i/2n) z(i-1/2n) = 2 X(1/2n 2 + ) (1/2n).


Therefore,






n
2
El E g(i-1,'/2) [z(i/2) z(i-1/2"n ] 12
i=l

2n 2n 2n
1n.21/
= g(i-1/2) g (j-1/2 ").2(/2n) + 7 g(i-1/" 2) (1/2n)
i=1 j=1 i=l


The double sum converges to


X 20 0 g(t)g(s) dtds = X (Jg(t)dt) (J g(s)ds)


= x2 0 g(t)dt 2
10
The sum on the right converges to Xj g(t) -dt. We conclude

That if the integral J1 g(t)z(dt) exists as a norm integral

in L2, then

2 1 2 1 d2 1 2
S(0 g(t)z(dt)) = X[Ifo g(t)dt 2] +).Ol|g(t) dt -


1 2
l0 g(t)gdt.


= XfOIg(t)I2dt.

Now since g is continuous we need only show that z is of weak

bounded variation as a mapping into L2; then the norm integral

.. O g(t)z(dt) belongs to L2, hence to L as well. By compu-
tations as above,

n n n
El I z(t ) Z(si) 12 S 2(t.i ) (tj -s j)
i=l i=l j= 1-

n
+ E X(t. s.)
i=l 1 1


< X2 + X,





92

since 0 s. s t. 1 for each i. .We conclude that z is of
1 1
weak bounded variation by Remark II.2.2. Since the simple
2n
functions E g (i-l/'2 n)
functions Ei g()i-ln n converge uniformly to

g on T, we conclude that
2n
o g(t)z(dt) = lim E g(i-1l/2n) [z(if2n) z(i-l2n)
i=l

in L2 and hence in Ll. The expressions for the mean and

variance now follow.

Motivated by this example, we can obviously define

sample path integrals under the following conditions:

(i) f:T is a continuous function and z(t) is a

process with almost every sample function of bounded varia-

tion.

(ii) f:T d is a function of bounded variation and z(t)

is a process with almost every sample function a continuous

function.

If either (i) or (ii) hold then by the formula for

integration by parts for Riemann-Stieltjes integrals (see

Rudin [26]) we have
1 1 1
(*1 .S f(t)z(dt, w) + 0 z(t,w)f(dt)= [z(*,w))f(.)]0

where both integrals are Riemann-Stieltjes integrals as in

Definition 11.2.4, and (*) holds for almost every w.

In case (i), random variables of the form
n
Z f(i-l/n) [z(i/n) z (i-1/n)]
i=l

converge to f(t)z(dt) for almost every w. In case (ii),
convege to0 (i







sums of the form
n
E z(i-l/n) [f(i/n) f(i-l/'n)]
i=l

converge to [0 z(t)f(dt) for almost every w. In both cases,

then, the integral exists as a measurable function on (P,F,P);

that is, the integral is a random variable.

In general, the existence and computation of the moments

of a sample path integral depend on the properties of the

functions Ez(t), Ez(t)z(s), and so on, as we saw in the case

of the integral with respect to a Poisson process.

A case of particular interest concerns random functions

whose sample functions are nonnegative and nondecreasing

functions. Since s < t implies that 0 < z(s) s z(t) almost

everywhere, we see that 0 s Ez(s) Ez(t) Ez(l). If we

suppose that z(1) belongs to LI, then it follows at once

that z is of weak bounded variation, since the mapping

t Ez(t) is nondecreasing. If f E L(z), then we know from

Theorem 1.5.8 that Esf (t)z(dt) = f(t)Ez(dt). The Poisson

integral discussed previously is a special case of this, since

Ez(t) = Xt.

We now discuss an interesting connection between norm

integrals and certain sample path integrals. Suppose that

z:T L is a random function of weak bounded variation, and

suppose that z is also measurable with respect to the product

measurable space (TxC., 1ExF). (E1xF denotes the smallest

a-algebra containing all rectangles AxB with A E E1 and B E F).

Shachtman [27] asserts that the following integration by parts

theorem holds, when p = 1.





94

4.1 Theorcm. If f:T -4 t is a function of bounded variation

and z:T L1 is product measurable and of weak bounded

variation, then f E L(z), and

SAf(t)z(dt) = jAz(t)mf(dt),
where the integral on the right is the sample path integral

with respect to the negative of the Lebesgue-Stieltjes

measure generated by f. A is taken to be an interval if

m is not countably additive, and a Borel set if m is count-
z z
ably additive.

The proof of this theorem (Shachtman [27]) discusses

only the case when A = T. That the theorem is false as

stated is trivial, since it fails even if z is a nonrandom

function, for example z(t) = t. For if f is constant then

mf 0. The question remains whether the formula

1 f(t)z(dt) = S z(t)mf(dt) + [zf]0
can be true under these hypotheses. We establish the follow-

ing result.


4.2 Theorem. Suppose f:T i is a continuous function of

bounded variation, and z:T L is product measurable and of

weak bounded variation, where 1 s p < m. Then
1 1 1
f0 f(t)z(dt) + J0 z(t)m(dt)= [zf],0
where the integral in the center is the sample path integral

of z with respect to the Borel-Stieltjes m on Z1 generated

by f.




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Vector Measures and Stochastic Integration By Franklin P. Witte A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1972

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UNIVERSITY OF FLORIDA illilllilllilllil 3 1262 08552 4642

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To Dot Whose patience and encouragement made this possible,

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ACKNOWLEDGEMENTS I would like to thank ray wife Dot for her understanding and encouragement throughout the past four years. I am indebted to each member of my committee; special thanks are due to Dr. James K. Brooks, who directed my research and guided my study of abstract analysis, and to Dr. Zoran R. Pop-Stojanovic, who guided my study of probability theory. Finally, I would like to thank Jean Sheffield for her excellent typing. Ill

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TABLE OF CONTENTS Page Acknowledgements iii Abstract v Introduction 1 Chapter I. Abstract Integration 5 II. Stieltjes Measures and Integrals 48 III. Stochastic Integration 76 References 105 Biographical Sketch 108 XV

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Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy VECTOR MEASURES AND STOCHASTIC INTEGRATION By Franklin P. Witte December, 1972 Chairman: J. K. Brooks Major Department: Department of Mathematics This dissertation investigates the stochastic integration of scalar-valued functions from the point of view of vector measure and integration theory. We make a detailed study of abstract integration theory in Chapters I and II, and then apply our results in Chapter III to show that certain kinds of stochastic integrals, previously defined by other means, are special cases of the general theory. In carrying out this program we prove extended forms of the classical convergence theorems for integrals. We also establish a generalization of the standard extension theorem for scalar measures generated by a left continuous function of bounded variation. The special case of measures in Hilbert space is discussed, and a corrected form of a theorem of Cramer is proved. In Chapter III we show that certain sample path integrals, the Wiener-Doob integral, and a general martingale integral are included in the abstract integration theory. We establish a general existence theorem for stochastic integrals with respect to a martingale in L , 1 < p < «>.

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INTRODUCT ION This dissertation concerns stochastic integration, a topic in the theory of stochastic processes. A stochastic process (random function) is a function from a linear interval T into an L space over a probability measure space. In its broadest terms, stochastic integration deals with linear transformations from a class of functions x to a class of stochastic process fxdz, which depend on a fixed process z. The functions x which are transformed may be sure (scalarvalued) functions, or more generally, random functions. Since randoia functions take their values in an L space they are vector-valued functions . Hence we are in the framework of the general theory of bilinear integration developed by Bartle [1] . The vector-valued function z generates a finitely additive measure on an appropriate ring or algebra of sets. When the integrand x is a sure function we are concerned with the integration of scalar functions by vector measures. When the integrand is another random function we have the full generality of the Bartle Theory, where both the integrand and the measure take values in Banach spaces X and Z respectively, and there is a continuous bilinear multiplication from XxZ into another Banach space Y, in which the integral takes its values .

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2 Until recently the field of stochastic integration appeared to be in much the same state as the area of real function theory was before the systematic introduction of measure theory. Several relatively unrelated kinds of stochastic integrals can be found throughout the literature on stochastic processes and their applications. Most of these integrals are defined as limits of approximating sums of the Riemann-Stieltjes-type. This approach, however, is not well-suited to stochastic integration in general, and certain difficulties may arise. With the exception of some work by Cramer [10] , the measure properties inherent in the stochastic integration scheme have not been exploited. There is currently a great deal of interest in unifying the field of stochastic integration using vector measure and integration theoiry. E. J. McShane, who has been investigating stochastic integration for several years, is presently writing a book which should contribute to this unification. The recent Symposium on Vector and Operator-valued Measures and Applications (Salt Lake City, August, 1972) was specifically concerned with stochastic integration, and attracted many of the known experts in both the fields of measure and integration theory (Brooks, Dinculeanu, lonescu Tulcea, Kelley, and Robertson) and stochastic processes and stochastic integration (Chatterji, Ito, Masani, and McShane) . A very recent unpublished paper by Metivier [23] , presented at the Symposium, deals with the problem of incorporating a rather general kind of stochastic integral within the framework of the Bartle integration theory.

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3 The purpose of this dissertation is more modest. We confine our attention to the case of scalar-valued integrands, and present a detailed investigation and synthesis of the appropriate vector measure and integration techniques, which are then used to exhibit certain kinds of stochastic integrals as special cases. In carrying out this program we make improvements in known results concerning the convergence of sequences of integrals, measures generated by vector functions, and existence of stochastic integrals. Chapter I develops the abstract integration theory of Bartle [1] for the special case at hand, namely, the integration of scalar functions by finitely additive vector measures. In this setting more is true than in the general Bartle theory. For example. Theorem 1.5.8 states roughly that the integral commutes with any bounded linear operator. This fact enables us to prove extended forms of the classical convergence theorems, and sketch a theory of L spaces. We discuss briefly the case of integration by countably additive vector measures, and introduce another integral developed by Gould [16] . This integral is defined by a net of Riemann-Stieltjestype sums, as opposed to the Lebesgue approach used in the development of the Bartle integral by means of simple functions. We show that these two integrals are equivalent a fact not discussed by Gould. Since a stochastic process is a function from an interval T of the line into an L space, as will be discussed in Chapter III, the stochastic integral of a scalar function f on T

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4 with respect to z can be thought of as a Stielt jes-type integral ["^^f (t) z (dt) in L_j. With this in mind, we investigate in Chapter II the properties of a Stieltjes measure m generated by a vector function z on T. We discuss the boundedness of m, and the question of the existence of a countably additive extension to the Borel sets of T. For a rather general class of Banach spaces we show in Theorem II. 3. 9 that the classical extension theorem for scalar measures on an algebra is valid. We also give a counterexample to show that the theorem fails to hold in the Banach space c . Finally we discuss the special case when z takes its valvies in a Hilbert space. We consider a theorem of Cramer [10] , which is incorrect as stated, and apply our results to establish a corrected version. Some implications of this theorem are then discussed. In Chapter III we first present some background information on probability and stochastic processes, and then motivate the study of stochastic integration. We show that the general integration theory of Chapters I and II includes certain sample path stochastic integrals, the Wiener-Doob integral, and a martingale stochastic integral. Several related results are also discussed.

PAGE 11

CHAPTER I Abstract Integration In this chapter we discuss the integration of scalar functions by finitely additive vector measures. Section 1 establishes the notation we will use, while Sections 2 and 3 list some basic properties of vector measures. In Section 4 we discuss the space of measurable functions and convergence in measure. Section 5 is concerned with integration theory proper, and in Section 5 we apply this theory to prove convergence theorems for integrals. Section 7 outlines part of the theory of L spaces in this setting. Countably additive measures are discussed briefly in Section 8. Finally, in Section 9, we compare a Riemann-type integral with the Lebesgue-type integral of Section 5, and establish the equivalence of these two approaches. 1. Notation . Throughout this dissertation the following terminology and notation will be used. N is the set of natural numbers {1,2,3,...} and N ={l,2,...,n}. The lowercase letters i,j,k,-t, and n will always denote elements of N. A is a finite subset of N. R denotes the real number field; we use the usual notation for intervals in R, so that (a,b) is an open interval, (a,b] a half-open interval, and so on.

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6 X and Y are Banach spaces over the scalar field §, which may be either the real or the complex field unless otherwise specified. |x| denotes the norm of an element X f X. X, is the unit ball of X, that is, X, = {x € X: |x| ^ 1} . B(X,Y) is the Banach space of bounded linear transformations from X to Y, with the usual topology of uniform convergence. X* = B(X, §), and X-|_* is the unit ball of X*. Recall that by definition, |x*| = sup lx*x| , ^1 while by the Hahn-Banach Theorem, lx| = sup lx*x| . ^1 Suppose that R and Z are families of subsets of a nonempty set S. R is a ring if R is closed under the formation of relative complements and finite unions. A ring R is a a-ring if R is closed un.der countable unions. S is an algebra if Z is a ring and Sen. An algebra Z is a aalgebra if Z is closed under countable unions. If S is any family of subsets of S, then a(S) denotes the smallest aalgebra containing g. If E € R, then tt(E,R) denotes the family of all partitions P = {E.:l^i^n} of E whose sets belong to R. When R is understood we write tt(E) . If (A^) is a sequence of subsets of S which are pairwise disjoint, we call (A.) a disjoint sequence. Whenever we write A = ZA^ instead of A = UA. for some set A and sequence (A. ) , we mean that (A.) is a disjoint sequence, and say that A is the sum of the A. 's. Similarly we write A + B instead of A U B, if A B = 0.

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7 Suppose that R is a ring and jn:P -* X is a set function, m is finitely additive if m(A + B) = m(A) + m(B) . m is countably additive if m(I'A.) = Em(A.) whenever (A.) is a sequence in R such that EA. € R. We make the convention that whenever we write m(A) it is assumed that A "belongs to the domain of m. A finitely additive set function defined on a ring or algebra will be called a measure. If m:R -• [0,00] is a set function, then m is monotone if m(A) ^ m(B) whenever A c B and A,B € R. m is subadditive if m(A U B) s: m(A) + m{B) for every A,B € R. m, ^ m means m, (E) < m(E) for every E e R. Standard references for results in measure and integration theory are Dinculeanu [12], Dunford and Schwartz [15], and Halmos [17] . We adhere to the notation of Dunford and Schwartz unless otherwise noted. 2. Properties of Set Functions . In this section we discuss some elementary properties of measures and their associated nonnegative set functions. R is a ring and m:R X is a measure unless otherwise noted. m is bounded if the set m(R) is a bounded subset of X, that is, sup_^-|m(E) I < 0= . The EfcR total variation v(m,E) of the measure m on a set E G R is defined by n v(m, E) = sup S lm(E. ) 1 . n(E) i=l ^ It is well known (see Dinculeanu [12]) that the set function v(m) is finitely additive but may not be finite even though m is a bounded countably additive measure. As Theorem 2.2 (ii)

PAGE 14

8 infra shov/s , v (m) is bounded whenever m is a bounded scalar measure. For this reason v (in) plays an important role in the theory of integration with respect to a scalar measure m, since it is a nonnegative measure which dominates m. When m is a countably additive vector measure and v (m, E) < «> for each E 6 ft, Dinculeanu [12] has shown that most of the results from the scalar integration theory carxy over to the vector case. In general, however, the total variation may be infinite and hence of little value. Thus we must use a more delicate device known as a control measure for m (see Theorem 3-3 infra ) . We now introduce a set function m called the variation of m. For every E 6 R we define m(E) = sup lm(F) I . FcE F6R It is obvious that m is bounded if and only if m is bounded. The following Lemma will be used several times in this dissertation. 2.1 Lemma . If a,,..., a are scalars, then n E la. 1 ^ 4 sup I Z a. I . i=l ACN^ i€A In particular, if x^,...,x^ € X and x* € X*, then n E |x*x. 1 ^ 4 sup I Z x*x. I ^ 4|x*| sup | E x^ | , 1=1 ^ AcN^ i€A AcN^ i€A n and i S a.x. I ^ 4 sup | a . | sup | S x.]. 1=1 ^ ^ lii^n AcN i€A

PAGE 15

Proof . Let A-, , . . . , A^ denote the sets of integers i such that Re a, > 0, Re a. s; 0, Im a . > 0, and Im a . s respectively. Then n n n S |a.i i: I iRea^l + Z llma^| i=l i=l i=l = E Re a . E Re a . + E Im a . E Im a . = Re E a^ Re E a^ + Im E a^ Im E a . ^1 ^2 ^3 ^4 ^ <: 4 sup I E a. I , AcN^ i€A where we have used the elementary inequalities |a| i JRea] + lima] and JRea| , j Im a | s | a| for any a € §. The inequalities concerning the x*x.'s are then immediate. Since n n n I E a.x I = Ix* E a.x. I ^ sup | a . | E | x*x . | i=l ^ ^ i=l ^ ^ isii^n ^ i=l ^ for some x* 6 X-,*, the final inequality follows from the preceding one. D As a first application of this lemma we establish the properties of the variation of a vector measure. 2.2 Theorem . (i) in is nonnegative, monotone, and subadditive. (ii) If m is scalar-valued, then v (m) 5: 4m. (iii) |m(E) I €. m(E) s sup v(x*m,E) i. 4m(E), for E 6 R. ^1 Proof . (i) fii is clearly nonnegative and monotone by definition. If E,F, and G belong to R, and G c E U F, then

PAGE 16

10 G = (G n E) + (G n (F\E) ) , so that I m (G) I = I m (G E) + m (G n (F\E) ) 1 s: iin(G n E) I + |m(G n (f\E) ) ] s: fn(E) + ra(F) . Therefore m(E U F) s: ffi(E) + m(F) for every E,F € R . (ii) If m is scalar-valued and (E.) € tt(E), then n E lm(E.)l i 4 sup I T m(E.)| n 4 sup |m(F)I , i=l ACN„ i€A FcE Fea by Lemma 2.1. Thus v(m,E) s 4m (E) for every E e R. (iii) Suppose that E € R. It is obvious that |m(E)l i in(E) . Using (ii) we have m(E) = sup |m(F) 1 FcE F6R = sup sup |x*m(F) I X * FcE F€R i sup v(x*m,E) ^1 ^ 4 sup sup |x*m(F) | X * FcE F€ft = 4m(E) . D 2.3 Remark . By the principle of uniform boundedness (see Dunford and Schwartz [15]) any function with values in X is bounded if and only if it is weakly bounded. In particular, m is bounded if and only if x*m is bounded for every x* G X* . If R is a a-ring and m is countably additive, then each x*m is also countably additive. It is known (see Halmos [17])

PAGE 17

11 that a countably additive scalar measure on a g-ring is bounded. Hence m is also bounded. The Orlicz-Pettis Theorem (see Dunford and Schwartz [15]) states that, conversely, if x*m is countably additive for each x* ^ X* and R. is a a-ring, then m is countably additive. The hereditary ring )i generated by a ring R, is the family of all subsets of S which are contained in some element of R. If n:R [o,°»] is a monotone set function, then we can extend |i to Ji as follows. Define \x^(E) = inf ^(F), F2E FeR for every E 6 W. It is immediate that \x is a monotone extension of ^, and yx s 0. If )a is subadditive then so is li . To see this, suppose that G,H ^ M and e > is fixed. Choose E,F € R such that G c E, H c F, and n(E) < |ji (G) + e, \l{F) < [1 (H) + e. Then G U H c E U F and so U^{G U H) S ^(E U F) i ^i(E) + ^ (F) < la"*" (G) + |a"^ (H) +2e. We conclude that n (G U H) ^ |ji (G) + u (H) . In this way we can extend fn to li. A set A € )i is said to be an m-null set, or simply a null set when m is understood, if m(A) =0. We shall always identify the extension of m to !a by ffi unless there is a possibility of confusion. Note that A is a null set if and only if for every e > 0, there is an E ^ R, such that A c E and in(E) < e. 3. Strongly Bounded Measures . We now introduce a fundamental property of certain vector measures, and discuss some of its implications. R is a ring and m:R X is a measure unless otherwise noted.

PAGE 18

12 3.1 Definitions . (i) m is said to be strongly bounded (s-bounded) if for every disjoint sequence (E.) in ii, liiii in(E. ) = 0. This property was introduced by Rickart [25], and later used by Brooks and Jewett [6] to establish convergence theorems for vector measures. Brooks [4] showed that sboundedness is equivalent to the existence of a control measure. This in turn provides necessary and sufficient conditions for the existence of an extension of a countably additive measure from S to o {Y.) when r. is an algebra. These results are presented in Theorems 3.3 and 3.4 infra. (ii) A series Tx in X is unconditionally convergent if for every e > there is a £i such that AHA = implies e e that Is X . 1 < e . i6A The following result states some basic properties of an s-bounded measure. The proof is omitted. 3.2 Theorem . (Rickart [25]) (i) If m is s-bounded then m is bounded. (ii) The following statements are equivalent: (a) m is s-bounded. (b) For every disjoint sequence (E.) in R we have lim fn(E. ) = . (c) For every disjoint sequence (E.) in 9,, Em(E^) is unconditionally convergent.

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13 Suppose that IT* is a fsunily of measures on Ji. We say that the measures in tT\ are uniformly m-continuous, or uniformly continuous with respect to m, if for every e > there is a 6 > such that |u(A) j < e for every u f tTl if fh(A) < 6. When fU contains only one measure ^ we say that p is m-continuous. [j, and m are mutually continuous if each is continuous with respect to the other. The most important characterization of s-bounded measures is the following result due to Brooks [4] which we state without proof. 3.3 Theorem . (Brooks) m:R X is s-bounded if and only if there is a bounded nonnegative measure m on R such that (i) m is m-continuous . (ii) m s m on R. Moreover, m is countably additive if and only if m is countably additive. The measure m is called a control measure for m. (i) and (ii) show that m and m are mutually continuous; it follows that m and m have the same class of null sets. A fundamental problem in measure theory is to find conditions under which a countably additive measure on a ring R has a countably additive extension to the a-ring generated by R . Using the previous result of Brooks we can prove an extension theorem for countably additive s-bounded measures on an algebra. This subject has been investigated by many authors. Takahashi [28] introduced a kind of boundedness

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14 condition less natural than, but equivalent to, the concept of s-boundedness . His result in the case of an algebra is therefore included in the following theorem. A measure m:R — X is said to be T-bounded if for every e > there is an n ? N such that if E, , . . . , E are disjoint sets in R, then |m(E.)| < e for some i 6 N . To see that T-boundedness is equivalent to s-boundedness, we remark that if m is not s-bounded then there is a disjoint sequence (E.) and an e > such that |m{E.)| > £# i € N, so m cannot be T-bounded. This shows that if m is T-bounded then m is s-bounded. Conversely, suppose that m is s-bounded with control measure m as in Theorem 3.3. Since m is a bounded nonnegative measure there is a constant K such that n E m(E.) ^ K for every finite disjoint family (E.) • If m 1=1 ^ is not T-bounded, then there is an e > such that for every n € N there are disjoint sets E, „,..., E _ in R with m(E.^) >e, -^ in nn — m 1 s i S n. This clearly contradicts the boundedness of m, so m is T-bounded. Since m and m are mutually continuous, m is also T-bounded. We now state the extension theorem for measures on an algebra. 3.4 Theorem . (Brooks [5]) Suppose T. is an algebra and m:E X is countably additive and s-bounded. Then m has a unique countably additive extension to a (D . Remark. Conversely, if m has a countably additive extension to a (Z) , then m is s-bounded on Z.

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15 Proof . Let m be a control measure for m as in Theorem 3.3. Then m is countably additive and bounded on E . It is well known (see Halmos [17]) that a nonnegative, bounded, countably additive measure on an algebra has a unique extension to a bounded, countably additive measure on a (D = ^n • ^^ identify this extension by m. The symmetric difference of two sets A and B, denoted by A A B, is defined by A A B = (a\B) + (B\A) . We also have A A B = (A U B)\(A n B) . The elementary set relations (i) AACcAABUBAC, (ii) AUBACUDCAACUBAD, (iii) AnBACDDcAACUBAD, (iv) a\b a c\d caacubad, are straightforward to verify. Define d:i:, x Z-i -• [O,®) by d(A,B) = m(A A B) . Since m is sxibadditive, (i) shows that d satisfies the triangle inequality, and hence is a semi-metric for E, . Moreover, from the m-continuity of m on Z and from the inequalities m(A\B) s m(A A B) and |m(A) -m(B) | S |m(A\B) | + |m(B\A) 1 , it follows that m is a uniformly continuous mapping from Z into X . Since Z, is the smallest a-algebra containing Z, it is known that for every A G Z, there is a sequence (A ) in Z such that lim ni(A A A) = (see Halmos [17] ) . This fact shows that Z is a dense subset of Z, . If we identify sets A and B in Z, whenever d(A, B) = 0, then m is well-defined on Z, since m(A) = m(B) whenever d(A, B) = for A,B € Z. Since X is complete, m has a unique extension to a continuous mapping from Z, into X. We denote this extension by m.

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16 To see that m is finitely additive on E, , we remark that (ii) , (iii) , and (iv) show that the mappings (A,B) -AUB, A n B, and a\b respectively from T^^xy^-, E, are continuous. Suppose A,B € T.^ and A Ci B = 6. Let (A ) and (B ) be sequences i n n in E converging to A and B respectively. (ii) and (iv) imply that the mapping (A, B) A A B is also a continuous function, so lim d(A AB , A A B) = 0. By the continuity of m we have lim ni(A^ A B^) = m(A A B) = m(A + B) since A A B = A + B. But m(A A B ) =• m(A \b ) + m(B \a ) , and by the continuity . n n n n n n ^ -^ of (A, B) a\b we have lim m(A \B ) = m(A\B) = m(A) and ^' n n lim m(B \a ) = m(B) . Thus m(A + B) = m(A) + m(B) . Fina^-ly, to see that m is countably additive on T. it suffices to show that if A."^ then lim m(A.) = 0. Since m is countably additive, lim in(A.) = 0. This in turn implies that lim m(A.) = since m is continuous on the space (S-i/d) . Q We quote without proof a final result from the theory of vector measures. The following theorem is a generalization of the well-known Vitali-Hahn-Saks Theorem. 3.5 Theorem. (Brooks and Jewett [6]). Suppose R is a a-ring and V:fi? — [0,oo) is a bounded measure. If n :R, -* X is a vcontinuous measure for each n, and if lim u (E) exists for ^n each E € R/ then the n are uniformly v-continuous . 4. Convergence in Measure and the Space M(m) . This section introduces the concept of convergence in measure for sequences of functions. Using this concept we can define the class of

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17 functions v/ith which the integration theory of Section 5 v/ill be concerned. From now on E will denote an algebra and m:S -• X is a bounded measure. Then m is a nonnegative, monotone, subadditive, bounded set function, and, in fact, these are the only properties that are needed to obtain the results of this section. F(S) denotes the family of all scalar functions on S . If f € F(S) we use the abbreviation [|f| > e] for the set {s e S:|f(s)| > e}. 4.1 Definition . A sequence (f, ) in F(S) is Cauchy in measure if for every e > we have lim m[lf.-f.|>e]=0. 1,3-00 -» A sequence (f, ) in F(S) converges to f 6 F(S) in measure if for every e > we have lim m[|f, -fl > e] = 0. If (f-K.) converges to f in measure we write m lim f, = f. The following lemma gives a useful equivalent formulation of convergence in measure. 4.2 Lemma . Suppose f » fi^ € F(S), k e N. Then m lim f, = f if and only if there are sets A. 6 Z and a sequence of positive numbers e, converging to zero such that lim m(A, ) = and \\{s) f (s) 1 s Cj^ if s ;^ Aj^. Proof. Let (A. ) and (e, ) satisfy the stated properties, and suppose e,6 > are given. Choose n such that e-i^ < € and m (A^) < 6 if k s n. Since [j f^ fj > e] c [ | f^ f | > e^] = A^ if k 5 n, we conclude that m[|f, -f| >e]<6ifk2rn. Thus m-lim fn^=f

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Conversely, suppose ra lim f,, f. For f 6 F(S), Dunford and Schwartz [15] define the quantity Ijfll = inf (e + m[|f| > e]), e>0 which is finite since in is bounded. To see that limH f, f jj = 0, suppose, to the contreiry, that lim|jf, f|l > 0. Without loss of generality we may suppose that j! f, f|| > e for k € N and some e > 0. There is an n such that ra[jf, f| > e/2] < e/2 if k ^ n. Hence e < llfj^-fll ^ e/2+m[lfj^f] > e/2] < e, which is a contradiction. Therefore liml|f, fjl = 0. For each k C N choose e, > such that e, + fn[|f f| > e, ] < < ||f, f|| + 1/k, Then choose A, f S such that [|f, -f| > e-^] C A, and m(A, ) < fn[|f, f| > e, ] + 1/k. The sequences (A, ) and (e, ) evidently satisfy the properties stated in the lemma. Q If E c S, then X„ denotes the indicator function of E, that is, the function whose value is 1 for s f E and for • S J^ E. We now define an important class of functions. 4.3 Definition . A E-simple function is any function f € F(S) which can be represented in the form f = .^ ^i ^E,' 1=1 1 where a. € $, and the sets E. 6 Z are disjoint and satisfy S = ZE. . When S is understood, f is called a simple function. Note that every simple function f has a canonical representation with E. = [f = a.]. 4.4 Definition . f € F(S) is measurable if there is a sequence of simple functions converging to f in measure.

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19 For f 6 F(S) and E c S we define the oscillation of f on E, written 0(f,E), by 0(f,E) = sup lf(s) f(t) 1. s,tgE We have the following characterization of measurability . 4.5 Theorem . f is measurable if and only if for every e > there is a partition (E.) e tt(S) such that fti(E^) < e and 0(f,E.) < e for 2 s: i ^ n. Proof . Suppose (f, ) is a sequence of simple functions converging to f in measure. Let (A, ) and (e, ) satisfy the conditions of Lemma 4.2 for f and (f-,^) • Fix e > 0, and choose k such that m(A, ) < e and e, < e/3. Suppose that n ^ • ^ f , = "Z a. Xp ; define a partition of S by setting F, = A, i=l i and F. = E._Aa, for 2:si^n + l. Ifs,t€F. for some i > 1, then |f(s) -f(t) I i |f(s) f3^(s)l + |f3^(t) -f(t) | s2e/3. Hence 0(f,F.) < e for 2 ^ i s: n + 1, so (F.) is the desired 1 • 1 partition. Conversely, if the conditions of the theorem hold then choose a sequence of partitions P, for e = 1/k. If P-,^ = (E--. ) / set f , = S f (s., )X , X ^^^ Ik E^^ where s^^ € E., for ls:isn,. Ifs^^ E,, then |f-(s) f, (s)| ^ max 0(f,E.) ^ 1/k. Since mCE^,) < 1/k / ^Jiisin, the sequences (E^) and (1/k) satisfy the conditions of Lemma 4.2, and so m-lim f, = f. We conclude that f is measurable. G

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20 Let M(m) = M denote the space of all measurable functions in F(S). Since we have introduced a concept of convergence in F(S) we can discuss the topological properties of M. The following result from Dunford and Schwartz [15] lists some of the useful properties of the space M. We shall omit the proof. 4.6 Theorem . (Dunford and Schwartz) M is a closed linear algebra in F(S). If g:# — § is continuous, then the mapping f -* go f is a' continuous function from M into M. The following result will be very useful in connection with the Dominated Convergence Theorem. 4.7 Theorem . If (f, ) is a sequence of simple functions and m lim f, = f, then there is a sequence (g,) of simple functions such that mlim g, = f, and |g, | i 2|f| on S, for each k € N. Proof . Let (A. ) and (e, ) be sequences as in Lemma 4.2 for f and (f^) . Define <3-^{s) = fj^(s) if s ;^ A^ and If^^Cs) I > 2e^, and gT^(s) = otherwise, for each k. Then if s ^ A, and |f^(s)| > 2e^, we have Igj^(s) f (s) j < e^^. If s ^ A^ and |fj^(s) I i 2e^ then |f(s)| i If(s) -fj^(s) 1 + \f^is) 1 ^ 3e^. so |f(s) gT,(s) I ^ 3e^ if s ^ A, . By Lemma 4 . 2 we conclude that m lim g^ = f. If s € A^ or if |f^(s)| ^ 2e^, then g, (s) = so certainly lg^(s)| s; 2|f(s)|. If s j^ A^ and I fj^(s) 1 > 2e^, then

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21 If(s) I ^ !fj^(s)| If^(s) f(s) ^ lfk(s)l ^k ^ l/2lf3^(s)l = l/2lg^(s)|. Thus [g (s) I ^ 2] f (s) I for s € S and k 6 N. D 5. A Lebesgue-type Integral . We now present the standard results of integration theory. Many of these theorems were stated by Bartle [1] in his fundamental paper; they are included for completeness. Throughout this section E is an algebra and mrS -* X is a bounded measure. If f is a simple function with representation n Z a. X , we define the integral of f over E € E by i=l ^ ^i n r fdm = E a.m(E n E.) . The following result lists the basic properties of the integral for simple functions. The set function X (E) = J fdm is called the indefinite integral of f. 5.1 Theorem. (Bartle) (i) The integral of a simple function is independent of the function's representation. (ii) For a fixed set E £ T. the integral over E is a linear map from the linear space of simple functions into X. (iii) For a fixed simple function f the indefinite integral of f is a measure on Z. (iv) If f is a simple function bounded by a constant K on a set E, then IJgfdm] ^ 4Kffi(E) .

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22 (v) If U € B(X,Y) and f is a simple function, then for every E € Z, Jgfdum = U(Jgfdm) . Note that by (iv) the indefinite integral of a simple function is a bounded and m-continuous measure. If m is s-bounded, then so is the indefinite integral of a simple function. The Bartle integral is defined using sequences of simple functions. The next two theorems discuss the properties of these sequences which enable us to construct the general integral. Bartle proved only that (i) implies (ii) in Theorem 5.2. 5.2 Theorem . Suppose (f, ) is a sequence of simple functions that is Cauchy in measure. Then the following statements are equivalent: (i) The indefinite integrals of the f^^'s are uniformly m-continuous . (ii) lim r f, dm exists uniformly for E € S. Proof . Suppose (i) holds and e > is given. Choose 6 > such that E e S and rfi(E) < 6 implies that for each k € N, IJgfdmj < e. There is an n € N and sets A^. € I such that m(A^.) < 5 and |f^(s) f.(s)| ^ e/4m(S), if s ;^ A^. and i,j 2 n. Then for E € L and i,j ^ n, we have ^ [4e/4m(s)]m(E\A^.) + 2e < 3e,

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23 using (iii) and (iv) of Theorem 5..1. Since E was arbitrary, we conclude that the sequences (J f dm) are Cauchy uniformly for E € S. Since X is complete, (ii) holds. Conversely, if (ii) holds and e > is given, then there is an n € N such that for every E € Z, provided k s n. Since each indefinite integral is m-continuous, there is a 6 > such that if A € S and m(A) < 6 then IJa^K^I < e, for 1 i k ^ n. If m(A) < 6 and k ^ n, we also have ^ 2e. Therefore the indefinite integrals of the f -i^ ' s are uniformly m-continuous . Q 5.3 Theorem . Suppose {f^^) and (g-,^) are sequences of simple functions such that (i) m-lim(f^ " "^k^ ^ °' (ii) The indefinite integrals of the f t^ ' s and the 9,^ ' s are uniformly m-continuous. Then lim Jgf^^dm = lim J^g^dm uniformly for E € Z. Proof . Since h, = f , g, is a simple function for each k, and since for every E 6 S, Theorem 5.2 implies that lim J hL dm exists for every E e Z. We need only show that this limit is zero

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24 uniformly in E. Fix e > 0. There is a 6 > such that if A e T. and fn(A) < 6, then for each k e N, IJaV^I < e. Since m-lim h^, = 0, there is an n 6 N and sets A, 6 S such that m(A^) < 6 and |hj^(s) j < e/4m{S) if s ^ A^, provided k ^ n. Thus for E 6 S and k s n, hdml Je\^^1 ^ IJe\A3^\^"^1 -^ IJeOA^^ S [4e/4m(S) ]m(E\Aj^) + e ^ 2e . That is, lim J h dm = uniformly for E € S. G Following Bartle [1] we now define the general integral. 5.4 Definition . f G F(S) is integrable if there is a sequence (f, ) of simple functions such that (i) m-lim f, = f. (ii) The indefinite^ integrals of the f -.^ ' s are uniformly m-continuous . If f is integrable then any sequence (f, ) of simple functions satisfying (i) and (ii) of Definition 5.4 is said to determine f. Theorem 5.2 shows that if (fn^) determines f, then lim Jpf-i^dn^ exists uniformly in E. We denote this limit by the usual symbols Jgfdm or Jgf (s)m(ds) . Theorem 5.3 shows that the integral of f is independent of the determining sequence. Let L(m) denote the family of all functions that are integrable with respect to m. The following

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25 theorem lists some standard properties of the integral. We omit theproof. 5.5 Theorem . (i) For a fixed set E € 2, the integral over E is a linear mapping from the linear space L(m) into X. (ii) For a fixed function f £ L (m) , the indefinite integral of f is an m-continuous measure on E. Suppose f G M and there is a null set A such that sup y |f(s) I < CO. Then we say that f is essentially bounded, and define the essential supremum ||f|| of f by llfll = inf sup |f(s) 1 . fli(A)=0 S;^A The following standard result, whose proof refines that of Bartle, shows that every measurable and essentially bounded function is integrable. 5.6 Theorem . If f G M is essentially bounded, then f 6 L(m) and for every E € E we have . IJjjfdml ^ 4l|fllfn(E). Proof . Let K = Hfjl, and suppose (f, ) is a sequence of simple functions converging to f in measure. Define gv(s) = " f^(s) |fj^(s) I ^ K, ^ (K/lf3^(s)|)f^(s) |f3^(s)! > K, for each s € S and k € N. Then (g, ) is a sequence of simple functions , and 93^(3) f^(s) I =< fj^(s)l K Ifj^Cs)! > K fv(s) I ^ K,

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26 since \g^(s) -fj^(s)| = j f^^ (s) (l-K/j f^ (s) | ) | = |f^(s)l-K if |f, (s)| > K. Suppose e > 0. Since f is essentially bounded by K, there is a null set A such that |f(s)l ^ K + e/4 if s ^ A. Now if s ^ A and Ifj^(s) j > K + e/2, then since |f (s) I ^ K + e/4, we have |fj^(s) f{s) 1 ^ !fj^(s)l |f(s)l > K + e/2 K e/4 = e/4 . Therefore [k^ " f^l > e/2] = [|f^| > K + e/2] c A U [|fk-f| > e/4]. Finally, [|gj^-f| > e] c [Igj^-fj^l >3e/4] u [|fj^-f| > e/4] c A U [|f^ f| > e/4]. Since A is null and m-lim f, = f , we conclude that m-lim g, = f. Since the g-,^ ' s are uniformly bounded by K, Theorem 5.1 (iv) implies that for every k, IJ^g^dml ^ 4Kifi(E) for any E £ T,. This shows that the indefinite integrals of the g^^ ' s are uniformly m-continuous . It follows that f € L(m), and that for every E € S, IJgfdml = limlj^gj^dml ^ 4Km(E) . n Using Theorem 5.6 and the m-continuity of the indefinite integral of a function f 6 L (m) we have the following Corollary 5.7 Corollary . If f € L (m) , then the indefinite integral of f is a bounded measure on E.

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27 Proof . Since f is measurable, f is bounded except possibly on sets of arbitrarily small variation. Choose 6 > such that if m(E) < 6, then IJgfdml < 1. Choose E € E such that m(E) < 6 and f is bounded on S\E, say by K. For any A € Z, therefore i 4Km(S) + 1. G When E is a a-algebra and m is a countably additive vector measure, Bartle, Dunford and Schwartz [2] have shown that the integral satisfies the following property. If f 6 L(m) and U € B{X,Y), then f € L (Um) and for each E 6 E, JgfdUm = U(Jgfdm). We now extend this result to the case where m is a bounded measure on an algebra E. 5.8 Theorem . Suppose that U € B(X,Y) . If f € L(m) then f € L(Um) , and for every E € Z we have JgfdUm = U(Jgfdm) . Proof . Since |Um(E) | ^ lu| | m(E) | ^. it follows at once that |Um(E)| <: |u|m(E) for every E € Z. From this inequality we see that if m-lim f, = f, then Um lim f, = f as well. If f € L(m) and (f, ) is a sequence of simple functions determining f, then we need only show that the indefinite integrals of the fi^'s with respect to Um are uniformly Um-continuous . Since we have

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28 using Theorem 5.1 (v) , this follows immediately from Theorem 5.2 and the fact that lim J* f,dm exists uniformly for E e >:. a When m is a scalar measure, Dunford and Schwartz [15] have introduced the following definition of integr ability . f £ F(S) is integrable if there is a sequence (f, ) of simple functions converging to f in measure, such that lim Jg|f^ f. Idv(m) = 0. i,j-*co We shall show that the Bartle concept of integration and that of Dunford and Schwartz coincide when m is a bounded scalar measure. We state the following result without proof. 5.9 Theorem . (Dunford and Schwartz [15]) If m is a scalar measure and f is integrable (in the sense of Dunford and Schwartz) , then the total variation of the indefinite integral X(E) = Jgfdni is given by v(X,E) = Jg|f(s)lv(m,ds) . Using Theorem 5 . 9 we now prove the equivalence of the Bartle and the Dunford and Schwartz integration theories in the case of a scalar measure. 5.10 Theorem . Suppose m is a bounded scalar measure. Then f e L(m) if and only if f is integrable in the sense of Dunford and Schwartz. In this case the same sequence of simple functions determines f for both definitions; consequently the two integrals coincide.

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29 Proof . Suppose (f-^) is a sequence of simple functions. By Theorems 5.9 and 2.2 (ii) v/e have ^ 4 sup j J (f. f.)dm| FcE -^ for every E € Z. Suppose f f L(m) and (f, ) is a sequence determining f. Then by Theorem 5.2 the right-hand term in the inequality goes to zero uniformly for E C E as i/j -• 00. Hence (*) lim J*g|f^ f.ldv(m) = 0, i,j-oo and so f is integrable in the sense of Dunford and Schwartz. Conversely, if m lim f , = f and (*) holds, then the lefthand term of the inequality converges to zero uniformly for E€Easi,j-»oo. This shows that the sequences (J f,dm) are Cauchy uniformly in E, and hence by the completeness of X, they converge uniformly in E. By Theorem 5 . 2 we conclude that the indefinite integrals of the fi^' s are uniformly m-continuous, so f € L(m) . In either case, the same sequence of simple functions determines f for both definitions, so the two integrals coincide. Q We now prove some results concerning the domination of functions by integrable functions. 5.11 Lemma . Suppose (f, ) is a sequence of integrable functions, and g € L(m) . If |f | ^ |g| on S for each k, then |J*j,fj^dml ^ 4 sup IJpgdm] FcE F€S

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30 for each E 6 Z; consequently the indefinite integrals of the f, ' s are uniformly m-continuous . Proof . We use Theorem 5.8 witli U = x* £ X*, and also Theorems 5.10, 5.9, and 2.2 (ii) to compute ^1 ^ sup r I f |dv(x*m) ^ sup r„ I g I dv (x*m) X * ^ -^1 i 4 sup sup I J gdx*m| FGE X * = 4 sup I r gdrnj, FCE ^ FGE for any k € N. Since the indefinite integral of g is mcontinuous, it follows that the indefinite integrals of the f. 's are uniformly m-continuous. G 5.12 Theorem . If f € M, g € L (m) , and ] f 1 ^ [g] on S , then f € L (m) . Proof . Since f G M there is a sequence (f, ) of simple functions converging to f in measure. By Theorem 4.7 we may assume that |f, I ^ 2|fl on S for every k. Then |f^| ^ 2|g| on S, and 2g € L(m) . By Lemma 5.11 we conclude that the indefinite integrals of the fi^'s are uniformly m-continuous. Therefore f € L(m) . a 5.13 Corollary . Suppose f e M. Then f € L(m) if and only if I f I € L(m) . If (f,) is a sequence of simple functions deter-

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31 mining f, then (] f , | ) is a sequence of simple functions determining 1 f j . Proof . The first statement follows from Theorem 5.12. If f, is a simple function, then so is | f > I » ^^'^ ^Y Lemma 5.11, |Jg|f^|dm| ^ 4 sup IJpfj^dmj. FcE F€Z Thus the indefinite integrals of the ] f , | ' s are uniformly m-continuous , if the indefinite integrals of the f -i^ ' s are. Finally, since | | f, | | f | | ^ | f, f | , it follows that [1 Ifj^l |fl I > e] c [If^fl > e] . If (f^^) determines f, we conclude that (|f, |) determines |f|. n 6. Convergence Theorems . In this section we prove stronger forms of the convergence theorems for sequences of integrals stated by Bartle [1] . Suppose that f and (f ) are integrable functions. We say that f and (f, ) satisfy property B if lim Jgf^dm = Jgfdm uniformly for E € S. The main convergence theorem resembles the classical Vitali convergence theorem. 6.1 Theorem . Suppose f € F(S) and (f,) c L(m). Then f e L (m) and f, (f, ) satisfy property B if and only if (i) m lim f, = f . (ii) The indefinite integrals of the fi^'s are uniformly m-continuous .

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32 Proof . Suppose f € L (m) and f/Cfi^) satisfy property B. To prove that m lim f , = f we proceed as follows. By Theorem 5.13, g^ = | f f ^^ | ^ L (m) for each k 6 N. Let (h^.) be a sequence of simple functions determining g as in Definition 5.4. By Theorem 5.13, ( | h, . | ) is a sequence of nonnegative simple functions determining |g | = g, , so without loss of generality we may assume that h, . ^ for each j 6 N. By Theorem 5 . 8 we have g, 6 L(x*m) for every x* € X*, hence by Theorem 5.10 g, is integrable in the sense of Dunford and Schwartz. By Theorems 5.9 and 2.2 (ii) we have S^g^dv{K*m) = ;g|f3^-f|dv(x*m) ^ 4 sup lj'p(f3^-f)dx*ml FCE F6E ^ 4 sup |Jp(fj^-f)dm|, Fes if X* € X^*« Since f and (f, ) satisfy property B, it follows that given e > 0, there is an n such that if k ^ n and X* € X,*, then for every E € S, j£gj^dv(x*m) < e. Since (h, . ) determines g, , it follows in the same manner that for a fixed k s n, there is an t such that j ^ -t implies that for every E € E, Je1\j " gkldv(x*m) < €, provided x* € X,*. By omitting the first l-l of the functions h, . if necessary, we therefore conclude that for x* 6 X,*, E € Z, and k k n.

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33 .dv(x*m) < 2e . Since h, . is a nonnegative simple function the sets E, . = [h, . > y] belong to E for every y > 0. Moreover, Yv(x*m,F^.) 5 Jgh^.dv(x*m) < 2e, so v(x*m,Ej^j) ^ 2e/y, if X* C X * . By Theorem 2.2 (iii) we conclude that m (Ej^.) ^ sup v(x*m,Ej^.) ^ 2e/Y X^* Since mlim h, . = g, , there is an i € N and a set F, € S such that fn(F^) < e/y and l^j^^^^ ~ ^k^^^ ' s: Y if s ;^ F^. Therefore, if s j^ R . U F, , we have |f,^(s) f(s) I = g^(s) ^ |gj^(s) hj^^Cs)! + |\i(s)l ^ 2y , and in(E, . U F, ) ^ 3e/Y. Now if 6-, and ft > are given, then choose y > such that 2y < 6-, • Then choose e > so that 3e/Y < bjIt follows from the arguments above that there is an n 6 N and sets G, such that if k s n, then m(G,) < 5~ and |f, (s) -f(s)l 5 6-, if s j^ G, . We conclude that m lim f, = f . k To see that (ii) holds we note that the argument in the proof of (ii) implies (i) for Theorem 5.2 applies. Conversely, if (i) and (ii) hold, then for each k there is a simple function g, such that m[lg, -f, 1 > 1/k] < 1/k and for every E € E,

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34 As in the proof of Theorem 4.6, it follows that m lim g,^ = f. Suppose e > 0. Choose n such that 1/n < s. Then choose 6 > such that fn(E) < 6 implies ljj.^3^dnil < e and IJ^f^drnj < e, for 1 ^ k ^ n and i € N. Then if m(E) < 6, i 2e for every k, so the indefinite integrals of the g-,^ ' s are uniformly m-continuous . It follows that f € L(m). Since llE^k^-lE^^"^! ^ llE^^k gk)din| +llE(gk-f)dm| ^ lA + IJ^^^k ^)^l' we conclude that f and (f, ) satisfy property B. O We now state two important corollaries to Theorem 6.1 which are stronger forms of the results stated by Bartle [1] . 6.2 Corollary . (Dominated Convergence) Suppose (fi^) c L (m) and g € L(m) . If ] f, | ^ jg] on S for every k € N, then m lim f , = f if and only if f € L(m) and f, (f, ) satisfy property B. Proof. Suppose m lim f = f. By Lemma 5.11 the indefinite integrals of the fn^'s are uniformly m-continuous, so the conclusion follows from Theorem 6.1. The converse is immediate by this same theorem. Q 6.3 Corollary . (Bounded Convergence) Suppose (f, ) c M is a sequence of functions uniformly bounded on S. Then m-limf, = f if and only if f € L(m) and f, (f, ) satisfy property B.

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Proof . Since constant functions are integrable, the conclusion follows from Corollary 6.2. G 7, Lp Spaces . As an application cf the previous theory, we present a portion of L space theory. In this section Z is an algebra and m:I X is a bounded measure. Recall the following set of inequalities, which were used in the proof of Lemma 5.11. Iff€L(m) and E ^ T., then IJgfdml = sup IJgfdx ^m ^1* ^ sup r„| f i dv (x*m) £. 4 sup sup IJ fdx*m| FCE X^* = 4 sup I Jpfdml . FCE These terms are finite by Corollary 5.7. Let f € M. If 1 s; p < CO and jfj^ € L(m), define llfH = sup [f Jf!Pdv(x*m)]l/'P . Let L denote the space of all functions f 6 M such that lf|^ € L(m). Since F c E implies that IJpfdml s sup Jg|fldv(x*m), the inequalities above show that the two expressions s E6 up [;glf|Pdm]l/P ^^^ ^^p fj^,f|P^^(^*^)jl/p ^1

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36 are equivalent "norms" for the spaces L . We prefer to use the second of these expressions, since the propertJ.es of the scalar norms in L (x*m) are then available. It By Corollary 5.13, it follows that L (m) = L, . Moreover, since 1 ^ p ^ q < <» implies that |f| ^ 1 + [fj / we see that f G L° implies that f ^ L , by Theorem 5.12. Hence 1 p q From the elementary inequality [a + b]^ < 2-^(|a|^+|b|^), it follows that jf + gj^ < 2^(1 f|P + Igl^); if f and g are in if, then so is f + g, by Theorem 5.12. L is clearly closed under scalar multiplication, and hence is a linear space. It is easy to see that llafH = |all|fll for any ir ir scalar a. Moreover, llf + gl! = sup[Jg|f + glPdv(x*m)]^'^P ^1 ^ s up[(J lflPdv(x*m))^/P + (Jg|gl%v(x*m))^/P] ^1* ^ llfllp + llgllp. since llf|| = if f = 0, we conclude that |1 • |! is a semip P tO norm on L . P Suppose 1 < p < o= and 1/p + 1/q = 1. The Holder inequality shows that |fg| Ifl^ Igl"^ Ifllpllgllg pIUII^ . qllgll^ if f e L and q f L . By Theorem 5.12 we conclude that P q fg f L° . If x* 6 X, we have

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37 Jg|fgldv(x*m) < llfllpllgll-qd/p + 1/q). Hence Hfgll^^ < llf llp!!gllq. A sequence of functions (f, ) in L is said to converge K p in L° to a function f € L° if we have limllf, f j] =0. The p p k "p following convergence theorem is a consequence of Theorem 6.1 7.1 Th eorem . Suppose 1 < p < <» and (f, ) is a sequence in L . K p Then f g L° and limllf, f 1! =0 if and only if p " K "p (i) m lim f, = f . (ii) The indefinite integrals of the |f, |^'s are uniformly m-continuous . Proof . Suppose f ^ L° and lira||f, f\\ =0. If we set g = I f 1^ f]^ € L(m) , then the first part of the proof of Theorem 6.1 is easily adapted to show that m lim f. = f. Since the indefinite integral of |f| is m-continuous, and 11Ve"p ^ l^^k " ^"p "^ II^eI'p' ^"-^^ follows as in the proof of Lemma 5.2. Conversely, suppose that (i) and (ii) hold. By Theorem 4.6, 1 f -1. 1 converges to lf|" in measure, so by Theorem 6.1, f € L . Moreover, Theorem 4.6 implies that m lim| f, f | -^ = Since |fj^-f|P ^ 2^(1 fj^l^ + lf|^), we have IJ^Ifj^-flPdml ^ 2P-"2 ^^p |J^(lf^lP+|flP)dml, FCK Fsr by Lemma 5.11. Since the indefinite integrals of the |f, |"'s are uniformly m-continuous, we conclude that the indefinite integrals of the jf, -fj-^'s are also uniformly m-continuous.

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38 By Theorem 6.1, then. lim ;^|f^ flPdm = E' k uniformly for E € E. Finally, since llf^^ fljP^ 4 sup \^^\f^ f^dml, E€ S we have limjlf, f || = . D The inequalities at the beginning of this section show that convergence in L, is equivalent to property B. For if f, (f, ) belong to L? = L(m), then suplJgCf^^ f)dm| ^ llf^ f||^ ^ 4 sup|J^(f^-f)dml . E$ Zj Ec S Just as Corollary 6.2 followed from Theorem 6.1, we have; 7.2 Corollary . Suppose 1 ^ p < », (f, ) c L , g € L , and K p p jf I <. |g| on S for each k. Then m-lim f, = f if and only if f € L° and limHf^ f\\^ =0. Proof . Since jfi.!^ ^ 1 9,1 ^' th® indefinite integrals of the |f, 1^'s are uniformly m-continuous . The conclusions follow from Theorem 7.1. G 7.3 Corollary . Suppose 1 ^ p < ». The subspace of simple functions is dense in L . P Proof . Suppose f € L° . Since f € M there is a sequence (f^) IT of simple functions converging to f in measure, and by Theorem 4.7 we may assume that j f , | ^ 2|fl on S for each k. By Corollary 7 . 2 it follows that limjjf f]! = 0. Q

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39 8. Integration by Countably Additive Measures . Integration of scalar functions by a countably additive measure m: Z -» X, where r is a a-algebra, has been studied extensively by Bartle, Dunford and Schwartz [2], and also by Lewis [20], The Bartle, Dunford and Schwartz Theory uses a result concerning the equivalence of weak compactness in the space of scalar measures with the existence of a control measure (as in Theorem 3.3), and the Vitali-Hahn-Saks Theorem. By means of these powerful results the following definition of the integral yields a theory which coincides with that in Section 5 for countably additive measures . A sequence of functions (f, ) c F(S) is said to converge (pointwise) almost everywhere to a function f if there is a null set A such that lim fi^(s) = f(s) for every s j^ A. In this case we write ae lim f, = f . 8.1 Definition . f € F (S) is integrable if there is a sequence (f, ) of simple functions such that ae lim f, = f, and such that for every EST, the sequence (pEfv^"') converges in X. Pointwise almost everywhere convergence replaces convergence in measure in the countably additive situation by virtue of the nonnegative control measure m. Since m and m are mutually continuous and countably additive, the standard theorems on the relation between convergence in measure and almost everywhere convergence can be shown to hold. In particular, ae lim f. = f implies m lim f, = f. Moreover, since the indefinite integrals of the function f, in Definition 8.1 are m-continuous me. ures , the quirement that

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40 (f f, dm) converges for each EST, together with the VitaliHahn-Saks Theorem, show that the indefinite integrals of the f , ' s are uniformly m-continuous , as in Definition 5.4. 8.2 Remark . If f is finitely additive, T is a a-algebra, and m is s-bounded, so that there is a control measure m by Theorem 3.3, then we may apply Theorem 3.5. Thus if ( r f, dm) converges for every E, we conclude that the indefinite integrals of the f-^'s are uniformly m-continuous . Convergence in measure must be retained however. Bartle, Dunford and Schwartz establish a one-sided form of Theorem 6.1: If (f,) C L(m), ae lim f^ = f, and the indefinite integrals of the f j, ' s slto uniformly m-continuous, then f 6 L(m) and lim J^^k^"^ ~ Ie'^^"^ ^'^^ E € FIn view of Theorem 6.1 and the Vitali-Hahn-Saks Theorem we have the following extension: 8.3 Theorem. Suppose (f^) c L(m) and ae lim f^ = f. Then f 6 L(m) and f, (f, ) satisfy property B if and only if lim r f, dm exists for each E € T. In the case of convergence in L we have by Theorem 7.1: 8.4 Theorem . Suppose 1 ^ p < <=, (f,) C L , and ae lim \= '^^ Then f € L° and limllf^^ fH = if and only if lim JEif]^.|^«^ exists for every E ^ T,. For future use we state the following useful approximation theorem for the countably additive case.

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41 8.5 Theorem . (Dunford and Schwartz [15]) Suppose 1 < p < », r is an algebra, and Z = o {Z ) . Then the space of E^o o o simple functions is dense in L . Lewis [20] has developed an integration theory for countably additive measures with values in a locally convex space X using the Pettis approach of employing linear functionals . This integration theory coincides with that presented above when X is a Banach space. 9. A Riemann-type Integral . Gould [16] introduced an integral for scalar functions, which in the case of bounded functions is defined as a limit of Riemann-Stieltjes-type sums. Although he calls his integral an improper integral, it is in fact equivalent to the "proper" Bartle integral when m is a bounded measure. This alternative approach to integration has been extended by McShane [22] to obtain a very general theory. We identify the Gould and the Bartle integrals as the (G) and the (B) integrals respectively. In this section E is an algebra and m:i: X is a bounded measure. If P and P' are two partitions in tt(S), we write P' ^ P to mean that each set in P' is a subset of a set in P; we say that P' is a refinement of P. If P 6 tt(S) and P = [E, ,...,E ], then for a function f € F(S) we introduce the symbol S(f,P) to indicate any Riemann-Stieltjes-type sum n Z f(s.)m(E.), where s. €E., l^i^n. Let B(S,E) = B(S) denote the family of all bounded measurable functions on S.

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42 ^•1 Definition . Suppose f € B(S).. f is integrable (G) if for any e > there is a P 6 Tr(S) such that if p' $ tt(S) and P' s P, then |s(f,P) S(f,P')| < e. We omit the proof of the following Theorem. 9.2 Theorem . Suppose f € B(S) . f is integrable (G) if and only if there is an x € X with the following property: For every c > there is a P € tt(S) such that if P' ^ tt(S) and P' 2: P, then ls(f,P')-x| 0. Choose k 6 N and A 6 E such that m(A) < e/4K and I |f(s) f^(s) I < e/4m(S) if s j^ A. Suppose f^ = ^g.^ a^ X^ . i Let F, = A and F. = E. Aa, 2 ^ i ^ I + 1. Then P = (F.) is 1 1 1-1 1 a partition of S and f, is constant over F„,...,F , . If P' = (B.) ^ P and if f, (s) = b. for s 6 B., then |S(f,P') -Jgf^dml = |_E f(s^)m(B^) -_E Jg fj^dm| s I I (f(s.) -b.)m(B.) I + I Z (f(s.)-b )m(B.) B^nA=0 111 B_^g;^ 1 1 X s: 4[c/4m(S) ]m(s\A) + 4Kfn(A) + 8Km(A) ^ 4e,

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43 where we have used Lemma 2.1. ,; Since |f, | < 2|fl, Corollary 6.3 shows that f and (f,) satisfy property B. There is then a k € N such that j r f , dm xl < e. The argument above shows that there is a partition P 6 tt(S) such that if P' ^ P then js(f,p') -x| s5e. We conclude that f is integrable (G) and that the (G) and (B) integrals coincide. Q. The vector x will now be denoted by f fdm. If E € S and f ^ B(S) , then fX„ is bounded and measurable by Theorem 4.6, and we write Je^^"^ = Js^^E^ • Suppose f 6 M(m) . The definition of measurability implies that f is bounded except perhaps on sets of arbitrarily small variation. Let B(f) denote the family of all sets in H on which m is bounded. It is easy to see that B(f) is a ring of sets. Since f is integrable on each set in B(f), the class 1(f) = [J fdm:E 6 B(f)} is nonvoid. Moreover, 1(f) is a net in X over the directed set B(f), where E ^ F means E c F. With this motivation we can define the integral for unbounded measurable functions. 9.4 Definition . f 6 M is integrable (G) if for every s > there is an E 6 B(f) such that if F € B(f) and E n F = 0, then jj fdm| < e. In view of the fact that 1(f) is a net we have the following result due to Gould [16] .

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44 9.5 Theorem . (Gould) f € M is integrable (G) if and only if there exists an x $ X having the follov/ing property: For every e > there is an E 6 B(f) such that if F e B(f) and E c F, then | J f dm xj < cProof . If such an x exists, then since B(f) is a ring and the indefinite integral of f is a measure on B(f), we have IJpfdnil ^ IJp+E^dni x| + jj^fdm xj , for every E,F € B(f) such that E fl F = 0. Fix -. > and choose E € B(f) such that if G € B(f) and E c G, then I r fdm x| < e/2. If F € B(f) and E F = 0, then E c E + F 6 B(f), and so IJpfdmj < e. Conversely, suppose f is integrable (G) . Definition 9.4 implies that 1(f) is a Cauchy net in X. For suppose e > is given. Choose E € B(f) such that if F € B(f) and EH F = 0, then I r fdm| < e. Now if F,G 6 B(f) and F,G ^ E, we have IJf^^ Jg^^I ^ UfXe^^"^! " IJgXe^'^I s 2e, where we have again used the additivity of the integral. Let X denote the limit of 1(f) . It is then immediate that x satisfies the requirement of the theorem. D From this result we see that when f is integrable (G) , then r_fdm = lim f^fdm . ^2 B(f) '^^

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45 Suppose now that f € L (m) . To see that f is integrable (G) , suppose e > 0, and choose 6 > such that if in (A) < 6, then 1 r fdm] < e. Since f is measurable, there is a set E € E such that f is bounded on E and fn(S\E) < 6. It follows that for any A ^ Y. such that A n E = 0, m(A) < 6, and so ] f fdm] < e. By Definition 9.4, f is integrable (G) . To see that integrability (G) implies integrability (B) we need the following result established by Gould [16] . 9.6 Theorem . (Gould) If f is integrable (G) , then the indefinite integral of f is a bounded m-continuous measure on Z. Proof . Since 1(f) is a Cauchy net it is a bounded set, so the indefinite integral is bounded. Additivity follows from the additivity of the indefinite integral for bounded functions. To see that the indefinite integral is m-continuous, suppose e > 0. There is an E € B(f) such that if F € B(f) and E n F = 0, then IJ fdm| < e. Hence for A6S»AnE=0 implies that jf fdm| ^ e. On E, however, the indefinite integral is already m-continuous by Theorem 5.6, so there is a 6 > such that if ffi(A) < 6 then l/AnE"^^"^' ^ ®* '^^^ ^^ fft(A) < 6 , then We now prove a result that is the counterpart of Lemma 5.11.

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46 9.7 Lemma . If f,g g M, jfj < |g| on S, and g is integrable (G) , then for every A € S/ (*) !J;^fdml ^ 4 sup IJ^gdml . BCA B6S Thus f is integrable (G) . Proof. Suppose E £ B(f) and e > 0. The indefinite integral of f over E is an m-continuous measure by Theorem 5.5, so there is a 6 > such that if m{A) < 5, then I f fdmj < e. •J ATI E Since g is bounded except on sets of arbitrarily small variation, there is a set F ^ B(g) such that m(S\F) < 6. Since f and g are both bounded on E H F, we have IJ^^pfdml ^ 4 ^sup^ IJ^gdml i 4 sup I f^gdn^l » BcE •'^ by Lemma 5.11. Therefore, IJe^^I ^ IJeXf^^^I -^ IJehf^^"^! ^ e + 4 sup I r„gdm| . BcE "^^ Since e was arbitrary, we see that (*) holds for every A 6 B(f), and hence for every A € S. To see that (*) implies f is integrable (G) , suppose e > 0. Choose E € B(g) such that if F € B(g) and E n F = jZ5, then I J* gdml < e. For any B ^ T. with B fl E = 0, we have I r gdm| s e. Since B(g) c B(f), we conclude by (*) that if A € B(f) and A n E = 0, then I r,fdml ^ 4 sup I r„gdm| rs 4e. G iJa g^^ Ub B6S

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47 Suppose now that f is integrable (G) . Since f C M, there is a sequence (f, ) of simple functions converging to f in measure, and by Theorem 4 . 7 we may assume that |f, 1 < 2|f| on S. Since 2f is integrable (G) , Lemma 9.7 shows that the indefinite integrals of the f-i^'s are uniformly m-continuous, since by Theorem 9.6 the integral of f is m-continuous . By Definition 5.4, f g L(m). To see that the (G) and (B) integrals are equal, note that the definition of the (G) integral as the limit of I (f ) shows that there is an increasing sequence of sets E, € B(f) satisfying (G) Jgfdni = lim J^n^dm. Since the functions fT„ are measurable by Theorem 4.6, and since we can choose the E, 's so that m(S\E. ) < 1/k, it follows that m lim fX„ = f. Since f f L(m) and I fX_ I ^ I f ! on S for each k. Corollary 6.2 implies that (B) Jgfdm = lim Jg^^E. '^• The integrals therefore coincide.

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CHAPTER II Stieltjes Measures and Integrals In this chapter we specialize the integration theory of Chapter I to the case where the measure is generated by a vector-valued function on an interval of the real line. We shall call such measures Stieltjes measures, and the resulting integrals will be called Stieltjes integrals. By restricting the Banach space in which the vector function takes its values to be an L space over a probability measure space, we shall study stochastic integration. (Chapter III) . In the present chapter we shall investigate the question of boundedness of a Stieltjes measure, and the question of the existence of an extension to a countably additive measure on the Borel sets. Finally we discuss the important special case of Stieltjes measures with values in a Hilbert space. 1. Notation . Let T denote a closed, bounded interval in R, which for convenience we shall assiome is the unit interval [0,1]. Let S denote the family of all subintervals of T of the form [a,b) or [a,l], together with 0, where we shall always assvime that s: a ^ b ^ 1 . Finally, let E denote the family of all subsets of T which are of the form A=i:a., where (A.) is a finite disjoint family in g. Note that Z 48

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49 is an algebra of subsets of T, and that the a-algebra generated by T is the family of Borel sets in T, which is denoted by E-, • Suppose that X is a Banach space, and that z:T -. X is a function. Define a set function m:g X by setting m[a,b) = z(b) z(a) and in[a,l] = z(l) z(a). If A 6 § and A = y'h, where each A^ € §/ then it follows at once by the definition of m on § that m(A) = i;m(A. ). Suppose n k ^ now that A e T., and that Z A. and S B. are any two rei=l ^ j=l ^ presentations of A by finite svims of sets in g. Then the sets C. . = A. n B. belong to S for each i and j, and we have A. = Z C. ., 1 :S i s n, and B. = Z C^., 1 s: j :£ k . By the preceding remark, it follows that m(A^) = T. m(C^.), and m(B.) = E m(C^ .) , for each i and j . We therefore conclude that r m(A^) = Y. T. m(C^.) = Y. m(B.) . i i j j -^ We have just shown that if A € S and A = SA. is any representation of A by a finite sum of sets in §, then we can define m(A) = S m(A. ) / and this definition in independent of the representation of A. It is now immediate that m is finitely additive on S. In the sequel, we shall associate with each function z:T -« X the unique measure m defined as above on S. When we wish to emphasize that the measure is generated by z, we write m . We call such a measure m the Stieltjes measure generated by z.

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50 If m is a bounded Stieltjes measure on Z, then m determines a class L(m) of functions integrable in the sense of Chapter I. The S-simple functions will now be called step functions; the class of step functions coincides with the class of all S-simple functions. Every function that is the uniform limit of a sequence of step functions is obviously bounded and measurable with respect to any Stieltjes measure on S. Hence by Theorem 1.5.6, such functions are integrable, and the sequence of integrals of the step functions converges to the integral of the limit function. If m has a countably additive extension to S, , then Theorem 1.8.5 shows that the class of step functions is still dense in L(m) . 2. Bounded Stieltjes Measures . In this section z:T -» X is a function generating the Stieltjes measure m:i; X. We investigate conditions on z which insure that m is a bounded measure. By Remark 1.2.3, m is bounded if and only if x*m is bounded for each x* € X*. Now since x*m [a,b) = x*z(b) x*z(a) = m . [a,b) , for every interval z x*z [a,b) (or [a,l]) in §, it follows that x*m = m for every X* 6 X*. By Theorem 1.2.2 (ii) we know that the scalar measure m . is bounded if and only if it has bounded total x*z -^ variation. Recalling that each set in Z is a finite sum of sets in §, it follows that n n sup S |m(A.)| = sup S lm(A.)|, (A^)cS i=l ^ (Ai)cE i=l where each (A.) is a finite disjoint family. That is, the total variation of any measure m on E is the same as the

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51 total variation of m restricted to §. Therefore the measure m ^ is bounded if and only if x*z ^ n n sup Z !m^^(A.)j = sup Z |x*z(b.) -x*z(a.) I
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52 we may restrict the partitions P to be from n (T, §) . To see this we introduce the concept of the norm of a partition P € Tf(T,S). If P = { A. : 1 ^ i ^ n] , where each A. is an interval [a.,t)-) (and A = [a ,1]), then the norm of P, denoted by I PJ , is defined as follows: [pj = max (b . -a.) • l^i^n ^ ^ We now state the above mentioned result. 2.3 Theorem . If m is a bounded measure and f ^ C (T) , then for every e > 0, there is a 6 > such that if P and P' belong to TT(T,g) and |p'| , |p| < 5, then JS (f,P') S(f,P) | ^ c. In particular, this holds if ] p| < 6 and p' s p. Proof . By the continuity of f, there is a 6 > such that if is-t| < 6 and s,t 6 T, then |f(s) f(t)| < c/4in(T). Now if P,P' 6 tt(T, S) and if |p1,|p'1 < 6/2, then n k |S(f,P') -S(f,P)l = I Z f(s.)m(A.) E f(t.)m(B.)l i=l ^ ^ j = l J -' = 1 E f (s.) Z m(A. n B.) Z f(t.) S m(A. HB.) | i=l ^ j = l ^ 3 j = i D i=i 13 n k = 1 E Z (f(s.) f (t.))m(A. n B.) I . i=l j=l J 1 J The only nonzero terras in this double sum occur when m(A. n B.) ^ 0; in this case A^ n B. 7^ 0, and hence Is. t . 1 < 6 . By Lemma 1.2.1 we conclude th at |S(f,P') -S(f,P)| < 4[c/4m(T)]m(T) = e. The final statement of the theorem is immediate, n

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53 Motivated by this result and the classical definition of the Rieip.ann-Stielt jes integral, we make the following definition. Recall that B(S) is the set of all bounded measurable functions on S. 2.4 Definition . Suppose miE -^ X is a measure and f f B(S) . Then f £ RS (m) if for every e > there is a 6 > such that whenever P,P' € tt(T,§) and |p|,|p'l < 6, then |s(f,P) S{f,P') I < €. Using this definition we could prove as in Theorem 1.9.2 that if f f RS (m) , then there is a vector x € X to which the sums S(f,P) converge as | p| converges to zero. When m is bounded. Theorem 2.2 shows that C (T) c RS (m) , and Theorem 1.9.3 and Definition 1.9.1 show that RS (m) c L(m) and that the integrals coincide. In fact, if m is any measure from "Z to X, and if C(T) c RS (m) , then m is bounded. This follows from Remark 1.2.3 and the following result. 2.5 Theorem. (Dunford [14]) If m is a scalar measure and C (T) c RS (m) , then m is bounded. Proof . First we remark that if (P ) is any sequence of partitions such that | P | converges to zero, then for every f 6 C(T), the sums S(f,P ) converge to the integral of f with respect to m. Suppose that m is not bounded. Since m is a scalar measure we have only to suppose that the total variation of m is infinite. It follows that there is a

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54 sequence (P ) in tt(T,S) such that JP j < 1/n and S |m{A. ) j > n for each n, where P = (A.); we may also assume that m(A.) ^0. If for each n we choose a set of pxDints s. 6 A., then by the preceding remark, lim S(f,P ) = f fdm for every f f-_ C (T) , where the sums S(f,P ) are formed using the points (s . ) chosen for each P . n Consider the mapping U :C(T) -* X defined by U (f) =S(f,P ), n e N. Each U is linear and bounded, and lim U (f) exists n n' for each f € C (T) . Since C(T) with the normllfll= sup |f(s)l is a Banach space, it follows by the principle of uniform boundedness that there is a constant K > such that (*) lS(f,P^)| < Kllfll, for every f € C (T) . To obtain a contradiction we shall construct a function f € C (T) such that Hf]! < 1, but for some k n € N, |S(f,P )| > K holds. Choose n such that Y. |m(A.)| >K, ^ i=l ^ where P = (A.) . Let s., 1 < i ^ k, be the sequence chosen for P above. Define f(s^) = m(A^)/| m(A^) | , 1 ^ i ^ k; f(t) = f(s,) for t in the interval [0,s^); f(t) = f(s^) for t in the interval [s, ,1]; and on each interval [s.,s._j_,) define f to be linear between f (s.) and f (Sj^. n) • Then f 6 C(T) , and we have llf|| = 1; however, k S(f,P„) = E f(s.)m(A.) k = E lm(A.) 1 > K. i=l Since this contradicts (*) , m is bounded. Q

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55 The following theorem summarizes the results concerning the boundedness of Stieltjes measures. 2.6 Theorem . Let mri: X be generated by z. The following statements are equivalent: (i) z is of weak bounded variation, (ii) m is bounded. (iii) C(T) C RS (m) . (iv) C (T) c RS (x*m) for every x* € X*. Proof . (i) » (ii) follows from Theorem 2.1. Theorem 2.3 shows that (ii) ^ (iii). We deduce from Theorems 1.5.8 and 2.3 that (iii) =* (iv) . Suppose that (iv) holds. Theorem 2.5 implies that x*m is bounded; hence, by Remark 1.2.3, m is bounded. G 3. Extensions of Countably Additive Measures . If z is a scalar function on T, then the classical extension theorem (see Halmos [17]) for the measure m states that m has a unique countably additive extension to I,, the Borel sets of T, if and only if z is left-continuous and of bounded (total) variation. When z is a vector function, we shall show that a similar theorem holds, provided that the Banach space X does not contain a copy of c . We need to introduce several preliminary concepts and results . A semi-ring § is a family of subsets of S such that (i) A,B € S implies that A n B € S; (ii) A,B € S and A = B implies that there is a finite family (C.) in g such that

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56 A=C, cc C...CC =B and C.\c. , f § for 2 f^ i ^ n 1 2 ^ 1 i"-'(see Halmos [17] or Dinculeanu [12] for information on semi-rings and measures on serai-rings) . If S is a semiring, then the smallest ring R containing §, which -we call the ring generated by S, is the family of all sums of finite disjoint subsets of §. An example of a semi-ring is just the class 8 of half-open intervals discussed in Section 1. In this case the ring generated by S is the algebra S. 3.1 Remark . Suppose that § is a semi-ring and that m:S --> X is a set function. It is known (see Dinculeanu [12]) that if m is finitely additive on §, then m has a unique finitely additive extension to R, the ring generated by S. Moreover, if m is countably additive on §, then its extension to R is countably additive on R as well. From now on, given a finitely additive set function on S, we shall assum.e that m has been extended to R, and shall denote the extension by m. When we refer to the variation in we always mean the variation of the extension, defined as in Chapter I, Section 2, by ffi(A) = sup lm(B) I . BCA B€R Note that if R is the ring generated by the semi-ring S, then we also have n ffi(A) = sup 1 E m(B.) I , (B^)=S i=l where (B.) denotes a finite disjoint family in §.

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57 If S is a topological space and A c S, then we denote the interior of A by int A and the closure of A by cl A. If S is a semi-ring in S and m: § -« X is a finitely additive set function, we make the following definition, with Remark 3 . 1 in mind . 3.2 Definition . Suppose that S is a compact space. Then m is regular if for every E € S and e > 0, there are sets A,B 6 S such that cl A c E c int B and m(B\A) < c. We nov/ prove a result due to Huneycutt [18] which generalizes the classical Alexandroff Theorem for regular scalar measures (see Dunford and Schwartz [15]). 3.3 Theorem . (Huneycutt) If S is a compact space, § is a semi-ring in S, and m:§ — X is finitely additive and regular, then m is countably additive on S. Proof . Let (E.) be a disjoint sequence in § such that E = HE. € S, and fix e > 0. By regularity there are sets A,B € S with cl A c E s int B and m(B\A) < e. There are also sequences of sets (A.) and (B . ) in S such that for each i, cl A. c E. c int B. and fn(B.\A.) < e/2'^. It follows XI 1 11 that for each n, n m[( U B.)\a] < 2e. For n n ( U B.)\A c [( U B.)\E] U [E\a] i=l ^ 1=1 ^ n = [ U (bAe. )] U [E\A], i=l ^ ^

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58 so n n in[( U B.)\a] s m[ U (B.\E )] + fn[E\Al i=l ^ i=l ^ ^ n i t: c/2^ + c < 2e. i=l 00 oo since cl A is compact and cl A c U E. c U int B., there is a k ? N such that Ac U B. . For n ^ k, therefore, i=l ^ n n n lm(A) Z in(E^) I ^ lin( U B^) m( U E^) 1 + i=l 1=1 i=l n + lm( U B.) m(A) I 1=1 n n n = lm[ U B.\( U E.)]| + |in[( U B.)\A]| 1=1 ^ i=l ^ 1=1 n n ^ ffi[ U (B.\E.)] + m[( U B.)\A] 1=1 ^ ^ 1=1 ^ " i ^ T. e/2 + 2e < 3e. 1=1 n Finally, |m(E) Zm(E.)l ^ |m(E) m(A) 1 + i=l n + |m(A) S m(E.) I 1=1 !S ffi(E\A) + 3e ^ 4e, 00 If n ^ k. We conclude that m(E) = Z m(E.); hence m is 1=1 countably additive on S. G As an immediate corollary of this result we have: 3.4 Corollary . (Huneycutt [19]) If m:Z X is a finitely additive measure such that lim m[a-h,a) = for every h-0+ a 6 (0,1], then m is countably additive on E.

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59 Proof . It: suffices to show that m is countably additive on S, and in viev/ of Theorem 3.3 we need only show that m is regular. Suppose that [a,b) g §. We may suppose that < a < b < 1, since the cases where a = or b = 1 are proved in a similar fashion. Choose h > such that < a-h < a 0, we can choose h > such that m[a-h,a) + m[b-h,b) < e. Therefore m is regular. r\ Vie now discuss an important class of Banach spaces characterized by Bessaga and Pelczynski [3] . The significance of this class for our purposes lies in the fact that it is precisely the class of range spaces for vector measures in which boundedness is equivalent to s-boundedness . Let c denote the Banach space of all scalar sequences X = (x ) such that lim X =0, with norm defined by ^ n n -^ [x| = sup |x I . If X is a Banach space and (x ) is a sequence in X, we say that (x ) is weakly Cauchy if the scalar sequences (x*x ) are Cauchy sequences for each X* 6 X* . We say that (x ) converges weakly to x € X if lim x*x = x*x for each x* . Recall that a Banach space is said to be weakly sequentially complete if every weak Cauchy sequence converges weakly to some element in X. A series Ex in X is said to be weakly unordered bounded if for each x* € X* we have sup I Z x*x. I < <=. A i6A ^

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60 Note that by Lemma 1.2.1, Zx is weakly unordered bounded n if and only if e|x*x. | < 0=, for each x* g X*. If a Banach space X contains no subspace isometrically isomorphic to c , then we write X ^ c ; otherwise v/e write X a c . We state the following important result without proof: 3.5 Theorem . (Bessaga and Pelczynski [3]) The following statements are equivalent: (i) X ;2^ c^. (ii) Every weakly unordered bounded series in X converges unconditionally. The following result was originally stated by Brooks and Walker [7] when X is weakly sequentially complete, and slightly extended to the present situation by Diestel [11] . 3.6 Corollary . If X ^ c and m:R X is any measure, then m is bounded if and only if m is s -bounded. Proof . By Theorem 1.3.2 an s-bounded measure is always bounded. Conversely, if m is bounded and (E.) is a disjoint sequence in R, then by Lemma 1.2.1 we have n T, |x*m(E.) I ^ 4|x*| sup I S m(E.)| i=l ACN^ i€A ^ ^ 4[x*| sup |m(E) I < ", E€R for every n e N and x* € X*. It follows that the series

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61 Zm(E.) is weakly unordered bounded. By Theorem 3.5, therefore, Zin(E.) converges unconditionally. By Theorem 1.3.2 we conclude that m is s-bounded. n 3.7 Example . To see that Theorem 3.5 fails to hold when X 2 c , consider the sequence of unit vectors e = (6 •) in o ^ n ni c , where 6 . is the Kronecker delta. The series He is o ni n weakly unordered bounded, since for any A c N, | E e. [ = 1, but this series does not converge in c . We now prove the final result that will be needed for the main theorem of this section. 3.8 Lemma . Suppose m:E X is a bounded measure such that lim m[a-h,a) = for every a€ (0,1]. IfX;^c, then for h-O"^ ° every a € (0,1] we have lim lft[a-h,a) = 0. h-O"^ Proof . Since m is monotone the limit above exists for each a 6 (0,1]. Suppose there is an a € (0,1] and a 6 > such that lim m[a-h,a) > 6. Let n, = 1. In view of Remark 3.1 h^O ^ there is an n_ > n, and a finite disjoint family of sets 2 1 n2 A^,...,A in g, such that | i: m(A.)l>6. We may assume 2 i=n,+l for convenience that A. = [a.,b.)# where a_ 6/2 . i=n^+l ^ ^2 i=n^+l

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62 Redefining b = b, so that A = [a ,b) , we have "2 "2 ^2 "2 I Z m(A^) 1 > 6/2. i=n,+l Now since ^[t) , a) > 5, we can repeat this argument over 2 the interval [b , a) to obtain an n^ > n_ and a finite ^2 J 2 disjoint family of sets A ,,..., A in S, such that 2 3 ^3 I E m(A.) I > 6/2. i=n2+l ^ By induction, therefore, we obtain an increasing sequence (n. ) and a disjoint sequence (A. ) in S such that for k € N, ^+1 I Z m(A.) ! > 6/2. i=n3^+l Define E, = Z A; then (E, ) is a disjoint sequence in Z, i=n3^+l and for every k 6 N, (*) Jm(Ej^) I > 6/2. Since X ^ c and m is bounded, m is s-bounded by Corollary 3.6. This contradicts (*) , so we conclude lim m[a-h,a)=0.n h-0 The following theorem is our main result on the extension of Stieltjes measures to countably additive Borel measures. 3.9 Theorem . Suppose X ^ c , and let Z^ be the a-algebra generated by S. The measure m :Z X has a unique countably additive extension to Z-, if and only if (i) z is left continuous on (0,1]. (ii) z is of weak bounded variation.

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63 If X 2 c , then there is a function z satisfying (i) and (ii) such that m has no countably additive extension to Z, . Proof . If m has a unique countably additive extension m z ' to Zt , then by Remark 1.2.3, m' is bounded, so m is z 1 z z also bounded. By Theorem 2.1 z is of weak bounded variation. Since m is countably additive on S, lim m (A.) = whenever (A.) is a sequence in E such that A.**^ 0. For any sequence a -. b , where b 6 (0,1], the sequence [a ,b) decreases to 0. Hence we have = lim m [a ,b) = lim [z(b) z(a )], so z is z • n L \ ' ' n left continuous on (0,1]. Conversely, if (i) and (ii) hold, then by Theorem 2.1, m is bounded. (i) implies lim m [a-h,a) = for every ^ h-.O'^ ^ a e (0,1]. By Lemma 3.8 we have lim. fn [a-h,a) = for h-.0 ^ a € (0,1]. Consequently, by Corollary 3.4, m is countably additive on S. Since X ^ c and m is bounded, m is sbounded by Corollary 3.6*. We conclude from Theorem 1.3.4 that m has a unique countably additive extension to E, . Now suppose that X 2 c . Let (e ) be the sequence of unit vectors in c defined in Example 3.7. Let e, n = 1 a = n n e, Z e. n ^ 2 i=2 Consider the function z:T — c defined by o ^ ^^^^ = J^ ^n^l/n-.l,l/n](^)

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64 for every t £ T. One can show that z is left continuous on (0,1]. Suppose that ^ s < t ^ 1. If s = 0, then since z(0) = 0, we have z(t) z(s) = a , provided that t e (l/n+l,l/n]. If s > 0, then for t g (l/n+l,l/n] and k s € (l/n+k+l,l/n+k], z(t) z(s) = a^ a^^^ = E e ^^, i=l provided k ^ 1. If k = then z(t) z(s) =0. Now if [a, ,b, ) , . . . , [a, ,b, ) are disjoint intervals in T, with a, < b-i ^ ••• ^ ^ < ^v» ^^*^ if ^1 ^ '-'' then there is a finite set n, ^ n_ :S ... ^ n , of integers such that ^2i z(b^) -z(a^) = E e., 1 :s i ^ k. ^=^2i-l If a, = 0, then z (b, ) z(a, ) = a . In either case, since k n E [z(b.) z(a.)] is a finite sum of the form Z c^e. for i=l ^ ^ i=l " ^ some n, where each eis or +1, we conclude that k I Z [2(b.) z(a.)]| ^ 1. 1=1 ^ ^ By Remark 2.2, z is of weak bounded variation. Now if m has an extension to a countably additive measure on Z-, # then m is s -bounded. Let A = ri/n+l,l/n) for each n. Then z n (A ) is a disjoint sequence in S, and m (A ) =z(l/n) -z(l/n+l) n z n e , n € N. Therefore Im (A„) | = 1 for every n, so m is not n z n z s-bounded. Hence m cannot have a countably additive extenz sion. G The requirement that z be left continuous is in a sense superfluous; if z(t~) exists for every t g (0,1], then we can define m[a,b) = 2(b") z(a~), (or m[0,b) =m(b~) -m(0)).

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65 Essentially this is just normalizing the function z to be left continuous. Moreover, as the following result shows, if z(t ) exists at each point of (0,1] and z is of weak bounded variation, then z determines a unique countably additive measure on E, . 3.10 Theorem . If z(t~) exists for every t $ (0,1] and if z is of weak bounded variation, then the function z' defined by z'(t) = z(t~), t € (0,1], and z'(0) = z(0) is left continuous on (0,1] and of weak bounded variation. Proof . Since by definition z' is left continuous on (0,1], it suffices to show that z' is of weak bounded variation. If [a, ,b, ) , . . . , [a ,b ) are disjoint intervals in T with a, < b, s . . . ^ a < b , then we can choose points af < a. 11 n n 11 (if a, = let a,' = 0) and bf < b., such that for each i, 1 111 ' |2(a.~) z(aM| < l/2n and |z(b.") z(b.')| < l/2n. Then n n _ n Is 2'(b.) -z'(a.)l ^ E |z(b.) -z(b.') I + s |z(aT) -z(a;)l i=l ^ ^ i=l ^ ^ i=l ^ ^ n + I S z(b^) -z(a()| i=l ^ 1 + ffi (T) . z By Remark 2.2 it follows that z' is of weak bounded variation. G Ifz:T-»Xis a function such that m has a countably additive extension to S, , then as we have already seen, z is left continuous. It follows that right-hand limits also exist at each point in [0,1). For suppose a ^ [0,1) and b^ a . Then the intervals [a,b ) decrease to [ a} 6 St / and

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66 so by the countable additivity of -m , m f a] = lira m fa.b ) = lim[z(b ) 2(a)]. Thus z(a ) exists. By Theorems 3.10 and 3.9, every function z with right and left-hand limits that is of weak bounded variation determines a unique countably additive measure on S-, . Conversely, it is easy to see that if m is a countably additive measure on S-, with values in X, then there is a function z with right and left-hand limits that is of weak bounded variation and satisfies 2(a~) = m[0,a) if a € (0,1], and z(a ) = ra[0,a] if a C [0,1) . If we normalize this function to be left continuous and satisfy z (0) 0, then we have established a one to one correspondence between the class of all countably additive measures on S-, and the family of all normalized functions z that are of weak bounded variation. This is a generalization of the classical theorem relating countably additive Borel measures and functions of bounded variation in the scalar case. 4. Measures in Hilbert Space . In this section we apply our previous results to sharpen a theorem of Cramer [10] , and discuss its consequences in integration theory. H is a Hilbert space with inner product {•,•), and z:T -• H is a -function which generates a measure m = m:I -• H. 5:Recall from Section 1 that the family g of all halfopen intervals in T is a semi-ring which generates S. If g :is any semi-ring, let gxS denote the class of all sets AxB, where A, B € g. It is not difficult to see that gxg is

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67 also a semi-ring. Elements of §xS are called rectangles. Consider the set function m, :§xS -• $ defined by F.j^(AxB) = . By the next lemma it will follow that m, is finitely additive on SxS/ and therefore by Remark 3.1, m, has a unique extension to a finitely additive measure on the algebra H generated by gxS. 3" is just the family of finite sums of rectangles in gxS. 4.1 Lemma. If AxB € SxS and AxB = A-|XB, + A„xB , then either A = A, = A^ and B = B, + B^ . or B = B, = Band Proof . If A-, = Ap, then clearly A = A-, = A. Suppose that B^ n B^ y^ ^. If b € B, n B„, then for any a € A, (a,b) € A-jXB, n 7i„xB„, which is a contradiction. Hence ^1 1^ ^2 = 0. Since B C B-j^ U B^ c B, E = B-, + B• If A, 7^ Ap, then there are a-,,a„ € A such that ^1 ^ ^1^^2 ^^^ ^2 ^ ^2^^!' ^^^^^ (a,,b,) and (a^/b-) £ AxB for any b^ € B^, i = 1,2, (a,,b2) and (a^/b,) € AxB. Since (a.j^,b2) € A-jXB,, B2 = B-,. Similarly, B, c B2/ so B = B, = B2 . Then as above we must have A, D Aj = and A = A, + A2 . D Now suppose that AxB = A,xB, + A2XB2, and that A=A, =A2 and B = B, + B2 . Then m-j^(AxB) = m^(Ax(B^ + B2)) = + = m^(Aj^xBj^) + m^(A2XB2) .

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68 It follows that m, is finitely additive on SxS* and so by Remark 3.1, we may assume that m, is a scalar measure on 3". 4.2 Remark . Suppose that (A^^) is a finite disjoint family in g. Then we have n n n I E in(A. ) 1^ = < Z m(A.), Z m(A. )> i=l " i=l " i=l " n n = 1:1: i=l j=l ^ 3 n n ^ E Sim (A.XA.) I . i=l j=l -^ ^ 3 We conclude that if m, is a bounded measure on 3", then m is a bounded measure on H, or equivalently z is of weak bounded variation. Since m, is a scalar measure. Theorem 1.2.2 (ii) implies that in^ i v(mj^) i 4m, on 3". It follows that in the definition of regularity. Definition 3.2, we may replace m^ by v(m,) . In what follows we shall use v (m, ) . The next lemma is the counterpart of Corollary 3.4. 4.3 Lemma . Suppose m, is bounded. Then m, is regular if (i) lim_ v(m^, [s,s^)x[t,t^)) = 0, s^ € (0,1], t,t^€T. s-»s o (ii) lim v(m^ , [s,s )x[t,t )) = 0, t € (0,1], s,s €T. t-»t~ X o o o o o Proof . Suppose [a,b)x[c,d) € SxS, and e > 0. By (i) there is a 6 > such that v (m, , [a 6,a) x [0,d) ) < e and v(m, , [b 6/b) X [0,d) ) < e. By (ii) there is an a > such

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69 that v(in, [0,b) X [d a,d) ) < e and v (m^, [0,b) x [c a, c) ) < c. Now let A = [a,b 6) x [c,d a) and B = [a 6,b) x [c u,d) . It follows that cl A c [a/b)x[c,d) c int B, and that b\a c [a 6,a)x[0,d) U [b 6/b) x [0, d) U [0,b) x [c a, c) U U [0,b) x [d a,d) , so v(m, ,b\a) ^ 4e. We conclude that m, is regular, since the same type of argument can be applied to any set in gxS. D We nov/ apply Lemma 4.3 to obtain a connection between the countable additivity of m, and the continuity properties of the generating function z. 4.4 Theorem . Suppose that m, is bounded. Then z(t ) and z (t ) exist for every t € (0,1) (with one-sided limits at and 1) . m, is countably additive on 0" if and only if z is left continuous . Proof . Suppose that lim sup |z(t) z(s) | > 6 > 0. Thai s , t-»t o there is a sequence s,
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70 Suppose that m, is countably additive on S"; then m, is continuous from above at 0. Since [t,t )x[t,t ) decreases to as t t~ , we conclude that ^ o = lim m. rt,t ) X [t,t ) . . 1 •• ' o ^ o t-*t o = lim_ lz(t) z(t^) 1 2 . t-t " o Therefore z(t~) = z (t ) . Conversely, suppose that z is left continuous . In view of Lemma 4.3 and Theorem 3.3 we need only show (*) lim_ v(m^, [s,s^)x[t,t^)) = 0, s^ € (0,1], t^t^ 6 T, s-»s o and similarly for t t~ as in Lemma 4.3, By Schwartz's inequality we have lm^[s,SQ)x[t,t^) I = | | ^ |z(s^) -z(s) I |z(t^) -z(t)|, and so be the ass\imption of left continuity of z we see that (**) lim_ m^[s,s )x[t,t^) =0 s-»s o for s € (0,1] and t,t € T. Since v(m^) is monotone, the limit (*) exists for every s € (0,1] and t,t € T. Suppose that for some choice of s ,t, and t we have o o lim_ v(mj_, [s,s^) X [t,t^) ) > 5, s-*s o for some £ > 0. Then there is a family of disjoint rectangles [a^,b^) X [Cj^,dj^) , 2 s. X S, n^, contained in [0,s^) x [t,t^) , such that S |m, [a.,b.)x[c. ,d.) | > 6. i=2 -^ ^ ^ X i

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71 Vie shall now show that we may assume b. < s for each i. Suppose that [a,b)x[c,d) is any rectangle in [0,s )x[t,t ) such that b = s , and linj^[a,b)x[c,d) | = a > e > 0, for some e > 0. By the continuity condition (**) we can select a b' such that a < b' < s and o |m^[b',s^) X [c,d) 1 a (a e) = e. Now if any of the b. = s , then the argument above shows that we can find new b. 's such that b. < s and 1 X o E |m, [a.,b.)x[c. ,d.) | > 6. i=2 Let 6, = min (s b . ) ; then v (m, , [s 6, ,s ) x [t/t ) ) > 6, X o --• ^' ,--« ox X O X O O ' Z^x^n^ so we can repeat the process above. By induction we obtain a disjoint sequence (A-xB.) of rectangles and an increasing sequence (n, ) of integers (set n, = 1) such that ^+1 S Im^ (A. xB.) I > 6, x=n3^+l contradicting the boundedness of m, . We conclude that (*) holds. The proof for t-'t" is similar to the above. Q o The theorem we are about to prove was essentially stated by Cramer [10] , but his statement and proof contain a gap which we now rectify. In our terminology, the statement of Cramer's result is as follows. Let z:T -» H be a function generating m and m, as above. If m, is a bounded measure.

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72 then m and m-, have countably additive extensions to Z, and 3"^ respectively, where ST, denotes the a-algebra generated by ST. Since even for scalar functions the measure m is not z countably additive unless z is left continuous, the theorem as stated is not correct. We now present the corrected version. 4.5 Theorem . If m. is bounded, then the measures m and m, have unique countably additive extensions to Z-, and 7, respectively if and only if z is left continuous. In this case we have (*) = m^(AxB) for every A and B in E, , where we identify the extensions by m and m, . Proof . If m, has a countably additive extension then z is left continuous by Theorem 4.4. Conversely, if z is left continuous, then by this same theorem m, is countably additive on JT. By the assumption of boundedness, m, has a unique extension to IT, (by the classical extension theorem or Theorem 1.3.4), and z is of weak bounded variation by Remark 4.2. By Theorem 3.9, m has a unique countably additive extension to To see that (*) holds, fix B € S, and define set functions H and X on Z, as follows: H(A) = mj_(AxB), X(A) = , for A € E-i . Since m and m, are countably additive, it follows that ^L and X are finitely additive and continuous from above at 0; thus \i and X are countably additive scalar measures.

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73 Since X (A) = u (A) for A € E, it follows by the uniqueness of extensions that X (A) = u (A) for A € Y.-^, or m^ (AXB) = for A € S-, and B € E fixed and B € S, fixed. Repeating this argument for A € Ei fixed and B € H-, gives the desired result, n The author is indebted to Professor Brooks for discussions on the previous material; in particular the proof of (*) in Theorem 4.5 is due to him. As an application of Theorem 4.5, we discuss the most important case, namely functions z with orthogonal increments. A function z:T H has orthogonal increments if = 0; m, therefore reduces to a nonnegative measure on S if we define m, (A) = = |m(A)| . It follows from orthogonality that m, is finitely additive, since m^{A + B) = = + + + = m(A) + m (B) . Since m, is nonnegative and finitely additive it is monotone, and so m, (A) ^ m, (T) for every A € E. By Theorem 4.5 we conclude that m and m, have unique extensions to E-, if and only if z is left continuous. Moreover, since m, is bounded. Theorem 4.4 implies that zCt"*") and z(t~) (z(0"*") and z(l~)) always exist. It follows that any function z with orthogonal

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74 increments determines countably additive measures m and m. if we normalize z to be left continuous. In fact, z is continuous except possibly at countably many points, since F(t) = m, [0,t) is a nondecreasing function and lz(t) -z(s)| = F(t) F(s) . Suppose now that z is any function with orthogonal increments, not necessarily left continuous, and z generates the measures m:i; H and miZ -* [0,co) as above. Since [m(A)] = sup |m(B) I = sup m, (B) = m^ (A) , BCA BCA BeS B€S it follows that convergence in measure with respect to m is equivalent to convergence in measure with respect to m, . Moreover, if (f, ) is a sequence of step functions on T, then the indefinite integrals of the fi^'s are uniformly m-continuous if and only if they are uniformly m, -continuous . Suppose n f = Z a. I is a step function; then 1=1 "• ^i ~ n n I Lfdmr = < S a.m(E HE.), S a.m(E HE.)) , ^ i=l ^ ^1=1"^ = _Z |a.|2m^(E n E.) = Jg|f|2dm3_, " using the orthogonality of the measure m. By the remarks above, and by Definition 1.5.4 and Theorem 1.7.1, we conclude that f € L(m) if and only if f € 'L°{m^) . Moreover, for step functions f and g it can be shown that = Iehf^^^i' and so by continuity this equality holds for every f and g in

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75 L(m) = Lp (m, ) . If z is left contimaous, so that rti and m, have countably additive extensions to Borel measures, then by Theorem 1.8.5 the class of step functions is dense in L (m) = L° (m, ) , and the results above still hold by continuity.

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CHAPTER III Stochastic Integration In this chapter we shall restrict the integration theory of the preceding chapters to the case where t?ie Banach space X is an L space over a probability measure space. Stieltjes integrals with respect to a function 2:T -• L are stochastic integrals in the sense that the integral is an element of L , or in probabilistic terms, a random variable. We show that the general integration theory of Chapters I and II includes certain stochastic integrals previously defined by other methods. In Section 1 we introduce and discuss the probabilistic concepts which are used. Section 2 consists mainly of background material on the theory of stochastic processes. In Section 3 we attempt to motivate the study of stochastic integration, and then discuss the general integration theory in the present context. The sample path stochastic integral is studied in Section 4; we compute some statistical properties of the integral in a simple special case, and discuss an integration by parts formula. The Wiener-Doob stochastic integral in L, is discussed briefly in Section 5. In Section 6 we deal with m.artingale stochastic integrals, and prove a general existence theorem. 76

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77 1, Probability Concepts and Notation . We say that (n,F) is a measurable space if is a nonvoid set (w is a generic element in in n) and F is a a-algebra of s\absets of Q. If there is a countably additive measure P defined on F with values in [0,1], and if P(fi) = 1, then we say that (n,F,P) is a probability measure space, and P is a probability measure. A probability measure space, or probability space, is therefore just a finite nonnegative measure space with the property that Cl has measure 1. Suppose that (Q,F) and (n',F') are two measurable spaces, and x:Q q' is a function. x is said to be measurable (relative to F and f') if the set [x € a'] = x~ (a') belongs to F for every a' € f'. The family of all sets [x € a'], a' € f'# is a a-algebra contained in F, and we denote it by F(x). Note that F(x) is the smallest a-algebra f" contained in F such that x is measurable relative to f" and f'. If P is a measure on F, then a measurable function x determines a measure p' on F ' by the relationship p'(a') = P[x € a'] for each a'€F'. IfPisa probability measure on F then p' is a probability measure on f' as well. In this case p' is called the distribution of x. When fi' = $ we shall always take f' to be the a-algebra of Borel sets in $. A measurable scalar-valued function on a probability space (Q,F,P) is called a random variable. As motivation for this term, suppose we have a system whose state can be described by a scalar variable x, and whose behavior is sxjbject to some sort of statistical variation. We imagine

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78 the numbers x corresponding to the (random) states of the system to be the values of a function x defined on some probability space. The assumption that x is a measurable function insures that the probability of each event x 6 B is defined, where B is any Borel set. This probability is just P[x € B] . Lp L (n,F,P), where 1 ^ p < oo, will denote the Banach space of all equivalence classes of random variables x such that e|x|^ h J^|x(uj)|Pp(dtJu) < ». L^ denotes the space of equivalence classes of essentially bounded random variables. It is known (see Dunford and Schwartz [15]) that for 1 ^ p < oo, Ly, is weakly sequentially complete. Since c is not weakly sequentially complete, it follows that L ^ c', hence Theorem II. 3. 9 applies to these p o ^^ spaces. The number e|x|P is called the p-th absolute moment "k of X. If k € N, then Ex is called the k-th moment of x. In particular, the first moment Ex is called the mean or expectation of x. It is well known from the theory of integration that the expectation operator E is a continuous linear functional on L, . If x € L-, then the number a^(x) = e|x-Ex1^ = e|x1^ |ex|^ is called the variance of x, and represents in some sense the dispersion of x about its mean value Ex. Standard references for results in probability theory are Loeve [21] and Chung [9] ; most of the material in this first section can be found in either of these works.

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79 2. Stochastic Processes . In this section we discuss the definition of a stochastic process (random function) as a mapping from a subset T of R into an L space over a probability space. In order to illustrate some of the important properties that a process may have, we discuss two classical examples, the Poisson process and the Wiener process. Let T be a subset of R whose elements are interpreted as points in time, and suppose we have a system whose observable state at time t 6 T is some scalar quantity x(t) . If the system is subject to influences of a random or statistical nature, then we are led to suppose that x(t) is a random variable for each t € T. Our mathematical model of the system is then a family of random variables x(t), t € T, which are defined on some underlying probability space (Q,F, P) . If we assxime that each x(t) belongs to L , so that the convergence properties of this space are available, then we have defined a mapping x from T into L . x is called a stochastic IT process or random function since it represents the behavior of a system subject to stochastic (random) influences. Since L is a space of (equivalence classes of) functions from n into f, we may also think of a random function x as a mapping from Txfi -• i . Then each uu € Q determines a function x(»,uj):T $; this function is called the sample function or sample path corresponding to uu. Every set A € F determines a set of sample functions [x(«,uj) :u) € A] , and we say that the probability of this set of sample functions is p if P(A) = p»

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80 A fundamental problem in the theory of random functions concerns the fact that we would like to discuss the probabilities of sets of sample functions which may not correspond to sets in F as above. For example, given any finite set of times t,,...,t and Borel sets B,,...,B , the set n [x(t^) € B^:l^i<:n] = n [x(t.) € B . ] i=l is in F. Even for countably many t.'s and B.'s such sets are in F. However, the set [x(t) € B:t 6 (a,b)] = n [x(t) € B] t€(a,b) clearly need not be measurable. Doob [13] discovered a reasonable way to skirt this issue. He was able to show that every random function x is equivalent to a random function y which behaves nicely in this regard, where equivalence means that for every t € T, P[x (t) = y (t) ] = 1. We shall always assvime for the sake of simplicity that we are dealing with such well-behaved processes. In order to discuss two important examples of stochastic processes we need to introduce a few additional concepts. Suppose that x(t), t 6 T, is a random function, and t,,...,t is a finite subset of T. Assume for simplicity that x(t) is real-valued for each t. Let R denote real Euclidean n-space, and let B denote the a-algebra of Borel sets in R . It can be shown that the family of all product sets B-,x...xB , where B. € B for 1 s: i ^ n, is a semi-ring which generates B . Consider the mapping uu -• [x(t, ,0)) , . . . ,x(t , u)) ] , which is a

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81 function from Q into R . Since each x(t.) is measurable, it follows that the set n [[x(tj_),...,x(t^)] €B^x...xB^] -_n [x(t^) e B^] belongs to F, for every product set B,x...xB . As in Section 1, the mapping uu [x(t, ,uj) , . . . /X (t ,(«)] determines a measure on the semi-ring of product sets, and by a standard extension theorem (see Halmos [17] ) , this measure extends to a probability measure P. . on B . The measure P. . is called the joint distribution of the random variables x (t, ) , . . . ,x(t ) , It seems reasonable, and in fact can be verified (see Loeve [21] ) , that a random function is characterized by the family of all its joint distributions P. . . 1 n Suppose that x(t) , t € T, is a random function. As in Section II. 4, we say that x(t„) x(t,) is an increment of the process. x(t) is said to have stationary increments if for every choice of t > and s,
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82 n known (see Loeve [21]) that if E rr |x(t.) x(s.)! < °' , i=l ^ ^ then we have n n E TT [X(t ) X(S )] = TT E[x(t.) X(S )]. i=l ^ ^ i=l ^ ^ Now we consider a classical random function called the Poisson process. Suppose we have a general situation where events occur in a random fashion that is in some sense uniform over T = [0,1] . Let x(t) denote the number of events which have occurred in the interval [0,t). It seems reasonable to assume that the random function x has the following properties : (i) Independent increments. The increment x(t) -x{s), t > s, represents the number of events occurring in [s,t), and we would expect that over disjoint intervals of time, the numbers of events occurring in those intervals are independent of each other. (ii) Stationary increments. The uniformity assumption means that over disjoint intervals of the same length, we expect that the distribution of occurrences is the same. (ill) x(0) ^ 0. If in addition we assume that there is a X > such that for small values of h > we have P[x(h) = 1] = Xh + o(h), and P[x(h) = 0] = 1 Xh + o(h) , then P[x(h) > 1] = o(h).

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83 and it can be shown (see Doob [13)) that for every s < t in T, we have P[x(t) x(s) = n] = e ^-X(t-s) ^ \"(ts)" nl for n = 0,1,2,... . This discrete probability distribution is called the Poisson distribution, and the relation above states that the random variable x(t) x(s) is distributed according to a Poisson distribution, with parameter X. The mean and variance of the increments are given by E[x(t) x(s)] = X(t -s) , and a^{x{t) -x(s)) =E[x(t) ^x(s)]2[E(x(t) -x(s))]^ = = \(t s) . The sample functions of a Poisson random function (except for a set of sample functions of measure zero) are monotones nondecr easing, nonnegativer integer-valued functions of t, with at most finitely many jumps of unit magnitude in T. This might be expected in view of the event-counting interpretation of x(t) . ; . A second important example of a random function is the Wiener process, which has been used extensively as a mathematical model for Brownian motion. Suppose we observe a particle undergoing Brownian motion, and restrict our attention to, say, the x coordinate of its position. Due to the random bombardment of the particle by the fluid's molecules, this X coordinate changes in an erratic fashion as time progresses. If we denote the value of x at time t s by

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84 x(t), then it seems reasonable to assume that x(t) is a random function which has the follov;ing properties: (i) Independent increments. The increment x(t) -x(s) represents a change in position along the x-axis over the interval [s,t). Over disjoint intervals of time these changes should be independent of each other due to the random character of the molecular bombardments. (ii) Stationary increments. Since we would assume that the fluid in which the particle is suspended is homogeneous, the distributions of change over two intervals of the same length should be the same. (iii) x(0) = 0, assuming we use the position of our initial observation as the center of the coordinate system. (iv) Normally (Gaussian) distributed increments, with mean zero and variance proportional to the length of the time interval. This follows from the assumption that the motion is random and continuous, and means that if a < b, then fb , 2 .^ 2 P[a a x(t) -x(s) is some constant, namely the variance over an interval of unit length (we have assumed t ^ so T = [0,co) ) . It can be shown that except for a set of sample functions of measure 0, the sample functions of a Wiener process are continuous, nowhere dif ferentiable, and of infinite variation over any finite interval. Although this last property runs

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85 counter to what intuition says the sample paths should look like, the Wiener process has played a very significant role in the application of probability theory to physics and engineering. In fact, the first stochastic integrals were defined by Wiener for scalar functions with respect to a Wiener process. A standard reference for results on stochastic processes is Doob [13] . Much of the material in this section can be found in his classic work. 3. Stochastic Integrals . We now discuss as example which motivates the theory of stochastic integration. Suppose we have a system consisting of a signal processing unit with an input signal z(t) and an output signal x(t) . Suppose that the effect of the processor can be described by a scalar function of two variables g(t,T) in the following way. A small change z(dT) = z(t + dr) z(t) in the input at time t produces a change in the output of the system at a time t ^ t given by g(t,T)z(dT). Moreover, suppose that the response of the system is linear, in the sense that small disturbances z (dx, ),..., z (dr ) produce a change in n the output given by Z g (t, x . ) z (dx . ) , for t ^ max x . . If i=l ^ ^ ^ we assume that g(t,x) = if t < x, then for a given input function z(t), we have formally that x(t) = J^^ g(t,x)z(dx) . As long as the functions g and z are sufficiently wellbehaved scalar functions, the definition of this integral poses no problem.

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86 Suppose, however, that the input z(t) is a random function, for example, a signal contaminated by random noise. Then the output will also be random, and we are faced with the problem of how to define the integral above so that it is a random variable for each t. If the function g is continuous in t for every t, and if almost every sample function of z is a function of bounded variation, then we could define the random output in a pointwise fashion as follows: (*) x(t,uj) = J"^ g(t,T)z(dT,uu) . Stochastic integrals defined in this way are called sample path integrals; although they are the most restrictive kind of integral, their study does provide motivation for the more powerful kinds of stochastic integrals. In general, this simple solution to the problem is not feasible, since even processes as well-behaved as the Wiener process have almost every sample function of unbounded variation. Using the integration theory of Chapters I and II when the Banach space X is an Lspace over a probability space, we obtain a very general definition of the stochastic integral. Suppose that z:T -* Lp is a random function, where 1 ^ p < OS. As in Chapter II we can define a measure m: Z -» Lp by setting m[a,b) = z (b) z (a) on g. For each A € I, m(A) is thus a random variable in Lp. The measure m is bounded if and only if z is of weak bounded variation, and by the form of Lp* this is equivalent, using Theorem I I. 2.1, to

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87 requiring that the scalar functions t -• E[z(t)x] are of bounded variation for every x € L , where 1/p + 1/q = 1 if 1 < p < <=, and q = » ifp= 1. Suppose that z is of weak bounded variation. Then z determines a class L(z) = L (m) of integrable scalar functions, and for f € L(z) we write J^f(t)z(dt) = J^f(t)m(dt) . We shall call such integrals stochastic Stieltjes integrals, or norm integrals for short, since they are defined as the limit of a sequence of integrals of step functions in the L norm. If JixH denotes the L norm (e|x|^) ^^ of an element X € L , then it is known (see Loeve [21] ) that 1 s; r :S p and It X € L implies that jjxll s: IJxH . This shows that L a L , cind that the canonical embedding U of L into L is continuous ^ pr p r By Theorem 1.5.8 it follows that if f € L{z) then f € L (U z) and That is, if f is integrable with respect to z as a mapping into L for 1 i p < 00, then f is integrable with respect to z as a mapping into L , 1 ^ r ^ p, and the integrals coincide. Moreover, it is clear that the same sequence of step functions determines the integrals of f in L and in L . By Theorem 1.5.8 we have E(J^f(t)z(dt))x = J^f(t)E[z(dt)x], for every x € L_, since as is well known, this is the form of linear functionals on Lp (see Dunford and Schwartz [15]).

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88 Thus E(J^f(t)z(dt)) = J^f(t)E[z(dt)] . Also El f^f (t) z (dt) I 2 = E [ J^f (t) z (dt) ] [J^f(T)z(d7) ] = J^f ( t) E [Z (dt) J^f(T)z(dT) ] . In general E[J,j,f(t)z(dt)]^ = J^f(t)E[z(dt) (J^f(T)z{dT))^~^], k— 1 since [J f(T)z(dT)] belongs to W/t,_-i if z:T L, . If we evaluate the scalar functions t E[z(t) (J^f(T)z(dT))^"^], then we can compute the k-th moment of the integral by performing a scalar integration. In practice this may be of some value. 4. Sample Path Integrals . Let T = [0,1] and g be a continuous function on T. Let z (t). be a Poisson process on T with parameter X. As described in Section 2, almost eveiry sample path z(',uu) is a monotone nondecreasing function, with z(0,tju) = 0, and finitely many jumps of unit magnitude. We can therefore define the Riemann-Stieltjes integral . jj g(t)z(dt,uj) for almost every m. By Theorem II. 2.3 it follows that jj g(t)z(dt,uj) = lim Z g (i-1/2'') [z (i/2'',aj) z (i-1/2'', uO ] X1-.00 i=l for almost every u). Since the summands g(i-l/2 ) [z(i/2 ) 'z(i-l/2")] are random variables it follows that the sample

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89 path integral J g(t)z(clt) is also a random variable. Moreover, we can evaluate the integral directly, due to the simple behavior of the sample functions. Fix o) and let n = z(l,o)). Let t,,...,t denote the n points of jump of o the sample function z (•,(«). For every n and i, the difference z(i/2 ,(d) z(i-l/2 ,(ju) is either 1 or 0, depending on whether one of the t. 's belongs to (i-1/2 ,i/2 ] or not. By the continuity of g we see that n 1 ° n g(t)z(dt,(ju) = E g(t.) . •^0 i=l ^ In general, denote by t . (tju) the point in T of the i-th jump of the sample function z(«,uj)7 that is, for t ^ t. (uj), z(t,a!) i i 1, while for t > t.(iD), z(t,uj) ^ i. Since [ujrt. (uj) < a] = [z(a) 2: i] £ F for each a € T, t. is a random variable for each i. Thus 1 z(l,u)) Jq g(t)z(dt,(ju) = r g(t^((ju)) . 1=1 That is, the integral of g with respect to the Poisson process z is the sum of a random number (z(l,uu)) of random variables of the form g (t . (uu) ) . ie: We can formally compute the mean of the integral as follows: we have . _ .. 2n 2^ -: E[ S g (i-1/2") [z(i/2")-z(i-l/2'')] = E g (i-1/2'') \ (1/2^) . i=l i=l 2^ , Since lim E g (i-l/2") X (1/2^) = X fi g(t)dt, and since the i=l JO 2^ functions Z g (i-1/2 )^r-;_i/2n 1/2^) converge uniformly to g on T, we would expect that E[Jq g(t)z(dt)] = XjJ g(t)dt.

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90 This will follow once it is established that the stochastic Stieltjes integral exists. Proceeding to the formal computation of the variance of the integral, we compute Ej S g(i-l/2") [z(i/2") z(i-l/2'')]r = i=l 2^ 2^^ = S S g(i-l/2'')g(j-l/2'')E[z(i/2")-z(i-l/2")] x i=l j=l X [z(j/2")-z(j-l/2'')] = E lg(i-l/2'')l^ E[z(i/2'')-z(i-l/2'')]^ + i=l 2^ + Z g(i-l/2'')g(j-l/2'')E[z(i/2'')-z(i-l/2'')] x X E[z(j/2'') z(j-l/2'')], since z has independent increments. To evaluate the second moments, we suppose that x is a Poisson random variable with parameter a. Then ^ OD _ n <» n— 1 ^2 ^ 2 -a a -a v^ Ex = Zne — r=ae Sn , -, x , -n=0 ^n=l ^'^-^^' ' '00 n-1 03 n-1 = ae [ae + e ] = a(a + 1) . Since z(i/2'^) z(i-l/2^) is a Poisson random variable with parameter X2~ , we conclude E[z(i/2'') -z(i-l/2'')]2 = X^(l/2'')^ + X(l/2"). Therefore,

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91 2 • , e| Z g(i-l/2'') [z(i/2'') z(i-l/2'')]r = i=l 2^ 2^ 2^ = I I g(i-l/2'')g(j-l/2'')X^(l/2")^+ 7; {g{i-i/2^) |2x(l/2"). i=l j=l i=l The double sum converges to \'^l\ Jo g(t)iTiT dtds = x2(jjg{t)dt) (jj ^TiTdF) = X^ljJ g(t)dtl2. 1 2 The sum on the right converges to XjQlg(t)| dt. We conclude that if the integral J^ g(t)z(dt) exists as a norm integral in L_, then a^(jj g(t)z(dt)) = X^[!jJ g(t)dt|2] +XjJ|g(t) | ^dt IXjJ g(t)dtl2 = XjJ|g(t)|2dt. Now since g is continuous we need only show that z is of weak bounded variation as a mapping into L^; then the norm integral r_ g(t)z(dt) belongs to L_, hence to L, as well. By computations as above, n _ n n „ e! S z(t.) -z(s.) |^= S Z X^(t. -s ) (t -s ) i=l ^ ^ i=l j=l 1 1 J J n + 2 X(t. s ) i=l ^ ^ i X^ + X,

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92 since ^ s. ^ t. ^1 for each i. We conclude that z is of weak bounded variation by Remark II. 2. 2. Since the simple 2^ functions H g(i-l/2 )1ri /-," • /on< converge uniformly to g on T, we conclude that 2^ Jq g(t)z(dt) = lim I g(i-l/2") [z(i/2") -z(i-l/2'')] i=l in Lp and hence in L, . The expressions for the mean and variance now follow. Motivated by this example, we can obviously define sample path integrals under the following conditions: (i) f:T $ is a continuous function and z(t) is a process with almost every sample function of bounded variation. (ii) f:T -» $ is a function of bounded variation and z (t) is a process with almost every sample function a continuous function. If either (i) or (ii) hold then by the formula for integration by parts for Riemann-Stielt jes integrals (see Rudin [26]) we have (*1 Jq f(t)z(dt,u)) + jj z(t,uj)f(dt) = [z(.,(ju)f(.)]J. where both integrals are Riemann-Stielt jes integrals as in Definition II. 2.4, and (*) holds for almost every uu. In case (i) , random variables of the form n S f (i-l/n) [z (i/n) z (i-l/n) ] 1=1 converge to ["_ f(t)z(dt) for almost every uu. In case (ii) ,

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93 sums of the form n Z z(i-l/n) [f(i/n) f(i-l/n)] i=l converge to r„ z(t)f(dt) for almost every uj. In both cases, then, the integral exists as a measurable function on (Q,F, P); that is, the integral is a random variable. In general, the existence and computation of the moments of a sample path integral depend on the properties of the functions Ez (t) , Ez(t)z(s), and so on, as we saw in the case of the integral with respect to a Poisson process. A case of particular interest concerns random functions whose sample functions are nonnegative and nondecreasing functions. Since s < t implies that i z(s) ^ z (t) almost everywhere, we see that ^ Ez(s) ^ Ez(t) ^ Ez(l) . If we suppose that z(l) belongs to L, , then it follows at once that z is of weak bounded variation, since the mapping t Ez(t) is nondecreasing. If f € L(z) , then we know from Theorem 1.5.8 that Ejjf (t) z (dt) = jjf (t) Ez (dt) . The Poisson integral discussed previously is a special case of this, since Ez(t) = Xt. We now discuss an interesting connection between norm integrals and certain sample path integrals. Suppose that z:T -• L is a random function of weak bounded variation, and XT suppose that z is also measurable with respect to the product measurable space (Txfi, S-iXF) . (Z-,xF denotes the smallest a-algebra containing all rectangles AxB with A 6 S, and B 6 F) . Shachtman [27] asserts that the following integration by parts theorem holds, when p = 1.

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94 4.1 Theorem, If f:T -4 $ is a function of bounded variation and z:T -4 L, is product measurable and of weak bounded variation, then f € L(z) , and J^^f{t)z(dt) = J^z(t)m^(dt), where the integral on the right is the sample path integral with respect to the negative of the Lebesgue-Stielt jes measure generated by f . A is taken to be an interval if m is not countably additive, and a Borel set if m is countz -^ z ably additive. The proof of this theorem (Shachtman [27] ) discusses only the case when A = T. Tliat the theorem is false as stated is trivial, since it fails even if z is a nonrandom function, for example z(t) = t. For if f is constant then m^ = 0. The question remains whether the formula jj f(t)z(dt) = jj z(t)m^(dt) + [zf]J can be true under these hypotheses. We establish the following result. 4.2 Theorem . Suppose f:T $ is a continuous function of bounded variation, and z:T -» L is product measurable and of weak bounded variation, where 1 ^ p < <». Then jj f(t)z(dt) + jj z(t)m(dt) = [zf]J, where the integral in the center is the sample path integral of z with respect to the Borel-Stieltjes m on S-, generated by f.

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95 Proof . Since z is of weak bounded variation and f is continuous, f € L(z) . z is a bounded function into L_, so sup Elz(t)|P < o=. Since the function t E|z(t)lP is measurable by the Fubini Theorem (see Dunford and Schwartz [15]), we have by this same theorem, E(jJlz(t)|Pv(m,dt)) = jj E|z(t)lPv(m,dt) < =o. Therefore the function uj J | z (t, U)) | ^v (m,dt) is F-measurable and finite almost everywhere. Since the function t -• |t|-^ is convex, the Jess en inequality for integrals implies that for almost every tju, we have \vd^ jjlz(t,a;) lv(m,dt) jP ^ ^^^ lll^it,^) Pv(m,dt) . Therefore E|jJ|z(t,uu) lv{m,dt) jP <: v(m,T)P~^EjJ|z(t,a)) |Pv(m,dt) < ». We conclude that the mapping ^ -* [q z(t,U))m{dt) is a random variable in Lp. Now since z is of weak bounded variation, each function t -• Ez(t)x is bounded, for x € Lg. By an application of Fubini 's Theorem we conclude that jj E[z(t)x]m(dt) = E(Jq z(t)m(dt))x, where Jz(t)m(dt) denotes the sample path integral whose existence was proved above. Since f 6 L(z), we know by Theorem 1.5.8 that E(J*J f(t)z(dt))x= jj f(t)E[z(dt)x], for each x € Lq. Using the integration by parts theorem (see Rudin [26]) we have Jq f(t)E[z(dt)x] = [f(.)Ez(.)x]Jjj E[z(t)x]m(dt),

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96 for each x € L_. Therefore E(Jq f(t)z(dt))x = E[[fz]J]x E(jJ z(t)in(dt))x = E[[fz]J jj z(t)m(dt)]x for each x G L . We conclude that jj f(t)z(dt) + jj z(t)in(dt) = [fz]J as desired, where the second integral is the sample path integral of z in Lp. D 4, Processes with Orthogonal Increments . In Section II. 4 we defined the concept of orthogonal increments for a function z:T -» H, where H is a Hilbert space. In particular, then, when H = L2(n#F,P), the results of Section II. 4 apply. Recall that the inner product in L» is defined by
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97 continuous functions n[g(t+l/n) -g(t)]). Using the orthogonality of the increments of the Wiener process, he then showed by somewhat lengthy computation that jj g(t)z(dt) 1^ = jj|g(t)!2dt. Using this identity and the fact that functions v;ith a E!j bounded derivative are dense in L (m^ ) , Wiener then extends the definition of the integr^ll to all of L (m, ) . This process of defining stochastic integrals was simultaneously simplified and generalized by Doob [13] . He replaced the functions with bounded derivative by step functions, and defined the integral with respect to any jorocess z:T L_ with orthogonal increments. The relation e|jJ g(t)z(dt)|2 = jl|g(t)!2m^(dt), where m, is a nonnegative measure generated by z as in Section II. 4, and g is a step function, is central to the whole development. By the results of Section I I. 4 it follows that the Wiener-Doob integral in 1, is contained in our general stochastic integration theory. Using the relation (*) E(J^f(t)z(dt)) (Jgg(t)z(dt)) = J^^gf(t)iTt)m^(dt), which is valid for f,g € L(m) = L (m, ) and A,B G S, it follows that if we define a new processes x:T -> L„ by X(t) = Jq f(T)z(dT), where f € L(m), then x is a process with orthogonal increments. For if s, < t, ^ s < t , then we have E(x(t2) -X(S2)) (x(t^)-x(s^)) =E(Jg2f(t)z(dt)) ( J^^f (t) z (dt) ) = ;[s,,t,)n[s2,t2)|f(^)l\(^^) =°-

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98 More generally, if z is left .continuous, then m and z m, have countably additive extensions to Z, . Since the class of step functions is dense in L(z) by Theorem 1.8.3, relation (*) is still valid. It follov/s by this identity that the measure ij:E, L defined by M(A) = J^f(t)z(dt), A € r,, is a measure with orthogonal values. Stochastic integrals with respect to processes with orthogonal increments have been used extensively in applications of probability theory to engineering. They are of particular value when the associated scalar measure m, can be determined, as is the case when z is a Wiener process. The measure m, allows us to compute the second moment of the stochastic integral of any f in L(z), by simply computing 1 ? the scalar integral fnl^l dm,. 6, Martingale Integrals . In this section we shall introduce the concept of conditional expectation, and use it to define an important class of random functions, the martingales. Using a deep result of Burkholder [8] , we show that every martingale z:T L is of weak bounded variation, provided 1 < p < 00. Consequently every such random function determines a class of integrable functions containing all uniform limits of step functions. Since unbounded measurable functions may also be integrable with respect to a particular random function, our integral is somewhat more general in this respect than the martingale integral of Millar [24] . The Millar integral is not restricted to scalar-valued integrands, however.

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99 Suppose X is a random variable in L, , and F, C F is a a-algebra. The indefinite integral of x is a P-continuous, count ably additive measure on F, ; by the RadonNikodym Theorem there is an integrable random variable z which is measurable with respect to F, , unique up to a P-null set in F-, , and satisfies r zdP = r xdP Ja J a for every A € F, . The random variable z is denoted by E[x|f, ], and is called the conditional expectation of x with respect to F, . The operator E['|f, ] maps L, into L, , and can be shown to be a continuous, positive, linear contraction operator. Suppose that A is any set in F, . Then we have, for every B 6 Fj^, JgE[xX^lF^]dP= J^xI^dP = Jgl^EtxiF^ldP. It follows that E[xXj.If,] = X E[xJF,] almost everywhere. By linearity and continuity of the operator E[»|F-,], we conclude that for any function y that is measurable with respect to F, and such that E[xy|F,] exists, we have E[xy|F,] =yE[x|F-] almost everywhere. In particular, E[ylF, ] = yE[l|F,] = y almost everywhere, so E[»|f,] is a projection operator.

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100 Another useful property of the conditional expectation operator will now be established. Suppose that F„ is another a-algebra, and F c F, g F. Then . J^E(E[x|F2] iF^)dP = J^E[x|F2]dP for every A € F^, so we conclude that E[E[x|F2] 1f^] = E[x|F2] = E[E[x|f^] IF2], up to some null set in F„. Finally, since Ex = jQ^dP = J^E[xlF^]dP, we conclude that E[E[x|F-,]] = Ex. Let T = [0,1]. We now define the martingale property. 6.1 Definition . Suppose 1 i p ^ <= and zrT L is a random function. Suppose that (F(t):t € T) is a family of a-algebras contained in F such that F(s) c F(t) if s ^ t. Then 2 (t) is a martingale with respect to (F(t):t e T) if z(t) is measurable with respect to F(t) for each t € T, and if s < t, then (*) E[z(t)|F(s)] = z(s) almost everywhere. We say for short that (z(t),F(t)) is a martingale. If (z(t),F(t)) is a martingale and t, < t2 < . . . < t^ is any finite set of points in T, then (z (t^) , F (t^) : 1 s: i 5: n) is a martingale in the sense that (*) holds for s and t restricted to the set { t, , ,t }. In this section we shall consider only real-valued random functions and scalar functions. An important result concerning conditional expectations is the conditional Jess en inequality. Ifg:R-Risa convex

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101 function, if x € L,, and if g (x) 6 L,, then for every aalgebra F-, contained in F, we have g(E[x|FJ) ^E[g(x)|F,] almost everywhere. For a proof of this result see Chung [9].. Suppose that z:T -» Lp is a martingale with respect to (F(t):t € T) , Since t Itj-^ is a convex function for every p s 0, it follows that for s ^ t, i2(s) |P= lE[z(t)|F(s)]|P ^ E[!z(t)lPlF(s)], almost eveirywhere. Hence e|z(s)|^ ^ E|z(t)|^, using the property E[E[x|F]] = Ex of conditional expectations. We conclude that the family of p-th absolute moments of a martingale in L is a nondecreasing family. Since EJzCt) 1^ ^ e1z(1) I^ for every t, we see that z is a bounded function from T to L_. P To show that any martingale z is actually of weak bounded variation when 1 < p < », we need to introduce the concept of a martingale transform. . 6.2 Definition . Suppose that (z ,F ) is a martingale in L , and (a ) is a sequence of real numbers. We say that a sequence (x ) of random variables is the martingale transform of (z ) by (a ) if for every n, n x^= i; a.[z. -z. ,], where we set z„ = 0. If (x ) is the transform of (z ) , then it follows that (x , F ) is a martingale. For x is measurable with respect

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102 to F , since each z., 1 ^ i ^ n, is measurable with respect to F . Moreover, ^ n+k E[x ., F ] = r a.E[z. z. , F ] ^ n+k' n-" . , 1 *• 1 x-1 ' n 1=1 n+k n = T. a.E[z.-z. Jf]+ T. a.E[z.-z. Jf] ... 1^1 1-1' n . , 1 1 1-1' n i=n+l 1=1 n+k n = E a. [z z ] + Z a. [z. z. ,] i=n+l ^^ ^ i=l ^^ ^-^ = X almost everyv/here. The following theorem is due to Burkholder [8] . 6.3 Theorem . (Burkholder) For each p, 1 < p < <=, there is a constant K„ such that if (x ) is the transform of (z_,F_) p n n n by a sequence (a^.) » with |a | i. 1 for all n, then e1x^|P ^ KpE|z^|P, n = 1,2,... . The proof of this result, which depends on showing, for each p, the equicontinuity of the sequence of bounded linear operators : n T^(-) = a3_E[.lFj_] + E a^{E[.lF^] E[.|Fj^_^]}, will not be given. We can now state our final result. 6.4 Theorem. Suppose 1 < p < <» and z:T L is a martingale with respect to (F(t):t € T) . Then (i) z is of weak bounded variation. (ii) If f € L(z) then E(jJ f(t)z(dt)) = 0.

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103 (iii) If f € L(z) and we define x(t) = Jq f(T)z(dT), for t e T, then (x(t), F(t):t € T) is a martingale. Proo f. (i) Suppose that t^ < t< ... < t^^ is any finite family in T. Then (z(t.)# F(t.):l ^ i ^ n) is a martingale. Let a„ . = 1 and a„ . , =0 for 1 ^ i 5 n. Then 2i 2i-l El S z(t2^) -z(t2i_l) |P=El S a^[z(t^) -z(t^_^)]|P i=l i=l 5 KpE|z(t2j|P ^ KpElz{l)|P < «, by Theorem 6.3. By Remark II. 2. 2, it follows that z is of weak bounded variation. (ii) Since z (t) = E[z(l)|F(t)] for every t ^ T, we have Ez(t) = E[E[z(l) I F(t) ] ] = Ez(l) . Hence Em [a,b) = E [z (b) z (a) ] = z for every interval in T. It follows that for every step function f on T, eJ„ f(t)z(dt) = 0. Since the class of step functions is dense in L(z), (ii) holds. (iii) Since f € L(z), there is a sequence of step functions f, such that the indefinite integrals of the f -i^ ' s converge uniformly for A € S to the indefinite integral of f. Define Xj^(t) = Jq f^(T)z(dT) for each k and every t € T. Then lim x, (t) = x(t) for every t, the limit taking place in L . It suffices to show that each x^ is a martingale with respect to {F{t) , t € T) . For if this is the case, then by continuity of the conditional expec-

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104 tation operator, \ve have E[x(t)|p(s)] =limE[x,(t)|F(s)] k ^ = lim X, (s) = x(s) . k ^ Suppose then that f is a step function of the form n-1 '^ ill """iti-rV * Vtt„_3^,tj' where -^ t^ < t-^^ < ... < t^ = 1 . If t 6 T, then x{t) = Jq f(T)z(dT) k-1 = S a. [z (t^) 2 (t^_^)] + a^ [z (t) z (t^_^) ] , where t ^ [t , t^) . Nov/ suppose that s < t. Then k E[x(t)|F(s)] = E a.E[z(t.) z (t . t)|f(s)] i=l ^ ^ ^"-^ k S a. [z(s) -z(s)] +a.E[z(t.)-z(t. ,) !f(s)] i=j+l ^ 3 J J--L J-1 + r a^[z(t^) 2(t^_^)] i=l = aj[E[z(tj) -z(s) |f(s)] +E[z(s)-z(tj_j^) |f(s)]] j-1 + E a^[z(t^) z(t^_^)] i=l j-1 = a. [z(s) z(t. ,)]+ S a. [z(t )-z(t )] J -'~i=l = x(s), where s € [t._,,t.). We conclude that (x(t),F(t)) is a martingale. Q

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REFERENCES [1] R. G. Bartle, "A general bilinear vector integral," Studia Math. 15 (1956), 337-352. [2] R. G. Bartle, N. Dunford and J. T. Schwartz, "Weak compactness and vector measures," Canadian J. Math. 7 (1955), 289-305. [3] C. Bessaga and A. Pelczynski, "On bases and unconditional convergence of series in Banach spaces," Studia Math. 17 (1958), 151-164. [4] J. K. Brooks, "On the existence of a control measure for strongly bounded vector measures," Bull. Amer. Math. Soc. 77 (1971), 999-1001. [5] , "Contributions to the theory of finitely additive measures," (to appear). [6] J. K. Brooks and R. S. Jewett, "On finitely additive vector measures," Proc. Nat. Acad. Sci. 67 (1970), 1294-1298. [7] J. K. Brooks and H. S. Walker, "On strongly bounded vector measures and applications to extensions, decompositions and weak compactness," (to appear). [8] D. L. Burkholder, "Martingale Transforms, " Ann. Math. Stat. 37 (1966), 1494-1504. [9] K. L. Chung, A Course in Probability Theory , Harcourt, Brace and World, New York, 1968. [10] H. Cramer, "A contribution to the theory of stochastic processes," Proc. Second Berkeley Symp. (1951), 329-339. [11] J. Diestel, "Applications of weak compactness and bases to vector measures and vectorial integration, " Rev. Roum. Math. Pures Appl., (to appear). 105

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106 N. Dinculeanu, Vector Measures ^ Pergamon Press, Oxford, 1967. J. L. Doob, Stochastic Processes , John Wiley, New York, 1953. N. Dunford, "Uniformity in linear spaces," Trans. Amer. Math. Soc. 44 (1938), 305-356. N. Dunford and J. T. Schwartz, Linear Operators , Part I_, Interscience, New York, 1958. G. G. Gould, "Integration over vector-valued measures," Proc. London. Math. Soc. 15 (1965), 193-225. P. R. Halmos, Measure Theory , Van Nostrand, Princeton, 1956. J. E. Huneycutt, Jr., "Extensions of abstract valued set functions," Trans. Amer. Math. Soc. 141 (1969), 505-513. , "Regularity of set functions and functions of bounded variation on the line," Rev. Roum. Math. Pures Appl. 14 (1969), 1113-1119. D. R. Lewis, "Integration with respect to vector measures," Pacific J. Math. 33 (1970), 157-165, M. Loeve, Probability Theory , Third Edition, Van Nostrand, Princeton, 1963. E. J. McShane, "A Riemann-type integral that includes the Lebesgue-Stieltjes, Bochner, and stochastic integrals," Mem. Amer. Math. Soc. 88 (1969). M. Metivier, "Stochastic integral and vector-valued measure, " (to appear) . P. W. Millar, "Martingale integrals," Trans. Amer. Math. Soc. 133 (1968), 145-166. C. E. Rickart, "Decompositions of additive set functions," Dioke Math. J. 10 (1943), 653-665. W. Rudin, Principles of Mathematical Analysis , Second Edition, McGraw-Hill, New York, 1964. R. H. Shachtman, "The Pettis-Stielt jes (stochastic) integral," Rev. Roum. Math. Pures Appl. 15 (1970), 1039-1060.

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107 [28] M. Takahashi, "On topological-additive-group-valued measures," Japan Acad. Proc. 42 (1966), 330-334. [29] N. Wiener, Nonlinear Problems in Random Analysis , M. T. Press, Cambridge, 1958 .

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BIOGRAPHICAL SKETCH Franklin P. Witte was born on May 30, 1942, in Davenport, Iowa. He attended parochial schools in Peoria, Illinois, and Conception, Missouri. In 1960 he graduated from high school at St. John Vianney Seminary, Elkhorn, Nebraska. From 1960 to 1964 he served in the U.S. Navy. He received his B.S. in Mathematics from the University of Missouri, Kansas City, Missouri, in 1957. As an undergraduate, he met his future wife Dorothy Silveirman; they were married in January, 1968. He worked at Midwest Research Institute in Kansas City for one year before enrolling as a gr.aduate student at the University of Florida in 1968. 108

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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. / / .' Vi-^'-l^ , /j^CA. 'C _ Jazties K. Brook?^' 'chairman P^rofessor 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. 4: fC ^ Kim Assistant 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. Zoran R. Pop-$tojanovic Professor of Mathenyktics 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. a Stephen A. Saxon Assistant Professor of Mathematics

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I certify that I have read this study and that in my opinion it confomis to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. A k J AP-H ^ Arun K. Varma Associate 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. l/aAU [P (v J^es W. Dufty ^sistant Professor o:^ Ph'ysics This dissertation was submitted to the Department of Mathematics in the College of Arts and Sciences, and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1972 S^Vkfir^i Chairman, Department of Mathematics Dean, Graduate School

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e> V9. s.« ,y'J.