SETVALUED INTEGRALS
By
MICHAEL JOSEPH SOUSA
A DISSERTATION PRESENTED TO THE GRADUATE
SCHOOL OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985
ACKNOWLEDGMENTS
I wish to express my sincere thanks to my advisor Dr. James K.
Brooks for suggesting the topic of this study and for providing
invaluable guidance and insight. Without his help this study would not
have been possible. I would also like to thank Steve Davis for our many
helpful discussions.
I thank the Mathematics Department of the University of Florida for
their financial help and encouragement.
My appreciation also extends to my typist, Jenny Wrenn, whose
excellent work made this task more managable.
Finally, I wish to thank my wife Maureen, who was very patient with
me while I completed this work, and my mother Judy and father Francis,
who have always encouraged me to study mathematics.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS...........................................ii
ABSTRACT...................................................iv
CHAPTERS
I. INTRODUCTION ......................................1
II. OPERATIONS ON SETS..........................9...... .9
II.1 Algebra of Sets..................... ...9.. .9
11.2 Convergence in Topological Vector Spaces...16
11.3 Convergence in Metric Space................18
11.4 Summation of Sets..........................27
III. THE AINTEGRAL............................... ........... 37
III.1 Definitions and Basic Properties..........37
111.2 Existence of the SAintegral..............42
111.3 Existence of the Aintegral...............47
IV. RELATIONSHIPS BETWEEN THE AINTEGRAL
AND OTHER INTEGRALS ..............................55
IV.1 The Dunford Integral.......................55
IV.2 The Aumann Integral........................ 63
V. APPLICATION OF THE AINTEGRAL TO
DECOMPOSITION THEOREMS........................... 67
V.1 Scalalvalued Set Functions................. 68
V.2 Vectorvalued Set Functions.................77
V.3 Setvalued Set Functions ....................83
BIBLIOGRAPHY ..............................................91
BIOGRAPHICAL SKETCH .......................................93
Abstract of Dissertation Presented to the Graduate School of the
University of Florida in Parial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
SETVALUED INTEGRALS
By
MICHAEL JOSEPH SOUSA
December 1985
Chairman: James K. Brooks
Major Department: Mathematics
Given sets A and B, solution of the equation A + X = B is
studied. Necessary and sufficient conditions for its solution are given
in case the sets are finite cross products in Rn of bounded closed
intervals in R. Two definitions of unconditional convergence of series
of sets are shown to be equivalent when the sets are bounded subsets of
a Banach space. Recalling Rickart's definition of the Aintegral, the
integrability of a finitely additive set function p is considered.
When i has finite total variation and is complexvalued, or has cross
products of bounded closed intervals as values, then i is A
integrable. If a Banach valued function f is Dunford integrable with
respect to a complexvalued measure u, then the setvalued set
function flI is Aintegrable, and the two integrals agree. If F(s) is a
subset of Rn for all s e S, 4 is a nonnegative measure, and the set
function Fu is Aintegrable, then the Aintegral contains the Aumann
integral. Finally, YosidaHewitt decompositions are given for finitely
additive set functions whose values lie in a topological vector space, a
normed linear space, and the collection of cross products in R of
closed bounded intervals.
I. INTRODUCTION
In recent years the study of setvalued functions and setvalued
integrals has enjoyed an increasing popularity. Setvalued functions
have been applied in areas as diverse as mathematical economics and
numerical analysis. Nobel laureate Gerard Debreu boosted interest in
setvalued functions in the economic world through a sequence of papers
concerning the properties of "correspondences." In these papers (see,
for example, ref. 10) he considers a complete, totally afinite positive
measure space (A,E,p), where A is a set of agents, and E is the ofield
of coalitions of agents. Debreu defines a correspondence to be a
function from E to the power set P(S), where S is an ordered finite
dimensional real vector space. With this definition he treats such
problems as the integration of a correspondence and finding the
RadonNikodym derivative of a correspondence.
A much more down to earth application can be found in R.E. Moore's
book Interval Analysis [17]. In this book Moore considers the problem
of mimicking calculations with real numbers on a finite precision
computing machine. Since every such computation is subject to roundoff
or truncation error, it is important to track these errors in order to
assess the accuracy of the final output. Moore follows these errors by
calculating with "interval numbers." Formally, an interval number is
defined to be a closed interval of real numbers. Intuitively, an
interval number represents an exact number plus uncertainty; to say the
result of a calculation is the interval number [a,b] implies that the
true result lies between a and b.
Mathematicians have studied setvalued functions and integrals for
their own interest as well. In 1972, Z. Artstein [1] considered set
valued measures defined on a measurable space and having values in a
finitedimensional real vector space. When the measure is bounded and
nonatomic, it is convexvalued. He was also able to prove that when a
setvalued measure i with convex values is absolutely continuous with
respect to a finite nonnegative measure X, then i has a RadonNikodfm
derivative with respect to X. L. Drewnoski [11] generalizes the
discussion of correspondences by allowing the range to be the subsets of
a locally convex topological vector space X. He then provides theorems
for the unique extension of a correspondence from a ring R to the aring
generated by R, and provides VitaliHahnSaks and Nikodfm type theorems
for sequences of countably additive correspondences whose values are
bounded subsets of X. In 1979, W. Rupp [22] generalized the Riesz
representation to finitely additive and countably additive setvalued
set functions defined on an algebra with values in the power set of Rm
These recent studies do not imply that the theory of setvalued
functions is a new development. As early as 1922 E.H. Moore and H.L.
Smith [16] were studying the convergence of generalized sequences of
sets. In "A General Theory of Limits" they consider a collection of
sets of complex numbers, the collection being indexed by a set S on
which is defined a relation R. The relation R is taken to be
transitive, and R has the composition property; that is, for
any s1,s2 in S there is an s3 in S so that s3Rs1 and s3Rs2. Now a
generalized sequence of sets (Xs)seS is said to converge to a number a
S sS
if for every positive e there is a s so that sRs implies x al < E
for all x e X F. Hausdorff in 1927 [13] was also concerned with the
s a
convergence of sets. He considered nonempty subsets of a metric space
(S,d) and defined the lower distance between sets A and B, denoted
6(A,B), to be the infimum of d(a,b) taken over all a e A and b e B. He
then defined the socalled Hausdorff metric for sets by the formula
H(A,B) = max [sup 6(A,b), sup 6(B,a)].
bEB aEA
This definition allowed him to say the sequence (An)n=1 converges to A
if lim H(A ,A) = 0. Hausdorff also defined four other notions of a
n n
limit of sets and compared these limits to the metric limit.
The idea of integrating a setvalued function is also not new. In
1930, A. Kolmogoroff considered the integration of real setvalued
functions defined on a measurable space, but he required the integral to
be singlevalued. Likewise in 1935, G. Birkhoff, using a generalization
of Frechet's "relative integral ranges," defined an integration process
for functions whose values are subsets of a Banach space. Given a
positive measure space (S,E,u), a function f defined on S with values
that are subsets of a Banach space, and a countable decomposition
A = {E.} of S, the integral range of f relative to A is the convex
closure of the set
C f(Ei)i(Ei).
i=l
After showing that any two integral ranges have a nontrivial
intersection, Birkhoff says f is integrable if the intersection of the
integral ranges is a point. So, as with Kolmogoroff, Birkhoff is only
willing to consider singlevalued integrals.
C.E. Rickart [20] showed in 1942 that he was willing to consider a
setvalued integral. He wanted to integrate functions whose values
where subsets of a locally convex topological vector space. To employ
limits in defining his integral, Rickart generalized Hausdorff's metric
limit of sets. He did this by utilizing von Neumann's system A of
neighborhoods about the origin (see [18]). Now the limit of a sequence
of sets (An) n=is A if for any V in A there is an nv so that
A A + V and A EA + V
n n
whenever n > n Now for a set E in the measurable space (S,E), Rickart
v
says f is Aintegrable if there is a set I(f,E) so that for any V E A,
there is a countable decomposition A so that if a decomposition A is
finer than A then the sequence (f(E n E ): En A)n= has the property
that there is a finite set of natural numbers n1 so that
f(E n E ) E I(f,E) + V and I(f,E) c Yf(E n En) + V,
n n
for any finite 7t containing 1. With this definition, Rickart provides
an integral that is a substantial generalization of the integrals of
Kolmogoroff and R.S. Phillips. Note that Rickart's integral is defined
just in terms of the function to be integrated; we do not integrate a
function against a measure, as we do with other integrals.
An entirely different approach to integrating setvalued functions
was provided by R.J. Aumann in 1965 [2]. For Aumann's integral we
consider S to be the unit interval [0,1], X the Lebesque measure on S,
and F a function from S into the nonempty subsets of euclidean nspace
En. Let F be the set of all functions which are Lebesque integrable
over S and f(s) E F(s) for all s E S. Now he defines the integral of F
over S to be
fFdX = {ffdX: f e F }.
S S
After defining this integral, Aumann proves a generalization of
Lebesgue's dominated convergence theorem.
In 1968 J.K. Brooks [5] considered the problem of integrating a
Banachvalued function against a finitely additive setvalued set
function. The set function i is defined on an algebra E subsets of some
set S, and the range of p consists of convex bounded nonempty subsets of
a real Banach space. With this setting he defines the integral of a
Banachvalued function f over E E by generalizing the Dunford
integration process (see [12]). First we define the integral of simple
functions
f(s) = ai 1 (s)
1 E.
to be
ffdt = ai p(E n E ).
E
If f is not simple, then we seek a sequence of simple functions
(fn) such that fn f in pmeasure, the integrals of the fun's are
uniformly absolutely continuous with respect to the variation v(p), and
the integrals are equicontinuous with respect to v(). If such a
sequence exists we can define f unambiguously as
ff du = lim ff nd.
E n E
For this integral, Brooks proves analogs of the Vitali convergence
theorem and the bounded convergence theorem. Furthermore, he
generalizes this entire process by defining a weak integral after B.J.
Pettis [19]. Here a realvalued function f is said to be weakly p.
integrable if f is x pintegrable in the sense defined above for
*
any x X and for each E in E there is a convex set A X such that
x A = ffd(x p) for x X ,
E
where equality means that the Hausdorff distance between the sets is
zero. Under these conditions the weak integral of f over E is
(w)ff dj = A.
E
In the present work, our principal aim is to demonstrate a
relationship between the Rickart Aintegral and the YosidaHewitt
decomposition of a finitely additive set function. The YosidaHewitt
decomposition of the finitely additive set function p can be expressed
as
i(E) = C(E) + p(E),
where C is countably additive and p is "purely finitely additive." To
achieve this decomposition for setvalued set functions we must solve
equations of the form
A = B + X
for the set X. This is the topic of chapter II section 1. All theorems
in this section are original. In section 3 we define the Hausdorff
metric and give necessary and sufficient conditions for its
completeness. The completeness of this metric belongs to the realm of
mathematical folklore, but the author has been unable to locate a proof
in the literature. So, we give an original proof. All other theorems
in this section are original. Section 4 deals with series of sets, a
problem central to defining a countably additive setvalued measure.
Again all theorem are original.
In Chapter III we define and discuss the properties of Rickart's A
integral. All theorems and definitions in section 1, except Defintion
8, are due to Rickart. In sections 2 and 3 we provide existence
theorems for the Aintegral, all of which are original except for
Theorem 3.1, which is due to Rickart.
Chapter IV gives relationships between the Aintegral and the
Dunford integral, and between the Aintegral and the Aumann integral.
All theorems in this chapter are original.
Chapter V contains our main theorems. In section 1 we show that
the Aintegral of a finitely additive complex set function i of bounded
total variation is exactly the countably additive part of the Yosida
Hewitt decomposition of B. In section 2 we give similar decompositions
for set functions whose values lie in a topological vector space, and
for set functions with values in a normed linear space. Our main
results are in section 3; we provide a YosidaHewitt decomposition for
finitely additive set functions whose values are closed bounded
"rectangles" in Rn. All theorems in this chapter are original.
II. OPERATIONS ON SETS
In order to effectively implement the notion of a setvalued
measure or a setvalued integral, certain natural questions must be
addressed. How do we define additivity of a setvalued function? What
is the proper notion of convergence of sets? Also, can we define
countable additivity for setvalued functions in a reasonable way? This
chapter aims at answering these questions and providing a firm
foundation for the theorems of the sequel.
11.1. Algebra of Sets
Mathematicans often find it useful to add, multiply, divide, and
perform other binary operations on quantities in some sort of consistent
and logical fashion. Fortunately, many of the algebraic properties of
vector spaces pass over directly to collections of sets.
1. Definition. Let S be a set closed under the binary operation *, and
let A and B be nonempty subsets of S. Then
A B = {a b:a E A,b E B}.
2. Note. If we take S to be a vector space in Defintion 1, with scalar
field D, then the following vector space axioms hold ref. 3 (p. 359):
9
1. A + B = B + A,
2. A + (B + C) = (A + B) + C,
3. a(A + B) = aA + aB,
4. a(PA) = (ap)A,
5. 1A = A,
6. + A = A,
7. 0 A= 0,
where A,B,C are nonempty subsets of S, a,P are elements of c, and 0
represents both the origin of S and the additive identity of .
Unfortunately, the subsets of a vector space S do not have additive
inverses; i.e.
1. A A # 0,
unless A is a singleton set, and the distributive property
2. (a + p)A = aA + PA
holds for all nonnegative a,P if and only if A is convex.
In the sequel, our decomposition theorems will require us to solve
equations of the form
A = B + X,
for the set X given the sets A and B. Thus property 1. is particularly
unfortunate as it prevents us from solving this equation in general.
However, in certain special cases, a solution can be found.
3. Theorem. Let ai, b be real numbers such that ai < bi for
1 = 1,2. Then
[al,bl] + [a2,b2] = [aI + a2,b1 + b2].
Proof. Let z e [al,bl] + [a2 + b2]. Then there is an
x e [al, bl] and a y E [a2, b2J so that x + y = z. Since
a < x < b1 and a2 < y < b2, we find that
al + a2 < x + y < bl + b2'
So z e [al + a2,b + b2] and thus
[al,bl] + [ a2,b2] [al + a2,b1 + b2].
Now let z e [al + a2,b1 + b2], and define
w = (z al
 a2)(b + b2 al a2)l
a2)(bI + b2 1 2)
Note that 0 < w < 1, and so if we define
cI = w(b1 al) + al'
then cl [al,bl] and c2
c2 = w(b2 a2) + a2,
E [a2,b2]. Furthermore,
c1 + c2 = w(b1 + b2 al a2) + al + a2
= (z al a2) + al+ a2 = z.
Thus [al,bl] + [a2,b2]
2 [al + a2,b + b2].
4. Corollary. Let ai, bi be real numbers with a < bi for i = 1,2, and
suppose bI a > b2 a2. Then there is a set X such that
[al,bl] = [a2,b2] + X.
Proof. The inequality
bI al > b2 a2 implies
bl b2 > ai
 a2, so define X = [al a2,b1 b2].
Then Theorem 3
yields [al,bl] = [a2,b2] + X.
This theorem can be easily extended to "rectangles" in R :
5. Definition.
Let II2... ,I be closed intervals in R.
(1, I2"..., n) = {x e Rn:xI e Il,x2 E 12,...,xn E In}
We call (I ,I2,...,I) a rectangle in Rn
6. Theorem. Let I. = [ai,b.] and J. = [ci,d.i be bounded closed
intervals in R for i = 1,2,...,n. Then
Define
( ,..., ) + (Jl, ..,Jn) = (KI,...,Kn),
where K = [ai + ci,bi + di] for i = 1,2,. . ,n.
Proof. For i = 1,2,...,n, let Ii = [ai,bi],
Ji = [ci'd], and Ki = [ai + ci,bi + di]. Let
x E (I*,...,I ) + (Ji,...,Jn). This is true if and only if there
exists y e (Il,...,In) and z e (J ,...,Jn) such that x = y + z. But
this holds if and only if for any i = 1,2,...,n, there exist yi e Ii,
and zi E J such that xi = Yi + zi, i.e. xi E Ii + Ji. By Theorem 3,
for every i = 1,2,...,n, we find by Definition 5 that
x E (I1,...,*n) + (Jl,...,Jn) is equivalent to x e (K1,...,K). O
In the case of a general normed linear space X, rectangles cannot
be defined. But a nice algebraic structure may be defined by
considering the arithmetic of closed balls. Subtraction is still
problematic in that additive inverses do not exist in general, but
addition of closed balls extends the addition of closed intervals in R.
7. Definition. Let X be a normed linear space, x E X and r > 0. Then
the closed ball of radius r about x is denoted B(x;r) and is defined by
B(x;r) = {y E X: y xl < r}.
8. Lemma. Let X be a normed linear space with additive identity 0 and
let r and s be nonnegative real numbers. Then
(a) B(0;r) + B(0;s) = B(0;r + s),
(b) x + B(0;r) = B(x;r) for any x E X.
Note: by x + B(O;r) we mean {x} + B(O;r).
Proof. Part (a): Let z E B(O;r) + B(O;s). Then z = x + y where
xI < r and y <( s. Thus izl = Ix + y < Ixl + lyl < r + s, which
implies z e B(O;r + s).
Now let z E B(O;r + s). If (zi < r, then z = z + 0 implies
z E B(O;r) + B(0;s). So assume r < Izl < r + s and write
S( )+ ( z r)z
IZI IZI
z z
Then ( r = r and z z r)z = Iz r < s; hence
z z
z E B(0;r) + B(0;s). This proves (a).
Part (b): Let z e x + B(0;r), then z = x + y where ly < r.
So jz xI = Ix + y x = ly < r, which gives us z e B(x;r).
Let z E B(x;r), then (z x1 < r, so write z = x + (z x).
Thus z e x + B(0;r). D
9. Theorem Let X be a normed linear space, x and y in X and r and s
both nonnegative. Then
B(x;r) + B(y;s) = B(x + y;r + s).
Proof. Lemma 5 implies B(x;r) + B(y;s) = (x + B(0;r))
+ (y + B(0;s)) = (x + y) + (B(0;r) + B(0;s)) = (x + y) + B(0;r + s)
= B(x + y;r + s). O
10. Corollary. If X is a normed linear space and r > s > 0, then for
any x and y in X, there is a set Y E X such that
B(x;r) = B(y;s) + Y.
Proof. Since r > s > 0 we have r s > 0. Let Y = B(x y;r s); then
Theorem 9 yields the desired result.
11. Note. In Corollaries 4 and 10 we required the diameter of B to be
less than or equal to the diameter of A in order to solve
A = B+X
for X, where, in a normed space
diameter Y = sup {jal a21: al,a2 E Y}.
This sufficient condition is easily seen to be also necessary, as the
diameter of a sum of sets is at least as large as the diameter of any
individual summand. To see this, note that if C and D are arbitrary
sets in a normed space, then for any E > 0, there exist cl, c2 in C such
that
cl c21 > diameter C e.
For any d in D, cl + d and c2 + d are in C + D, so
diameter(C + D) > (cI + d) (c2 + d)l = Ic c21 > diameter C e.
Hence diameter (C + D) > diameter C.
11.2. Convergence in Topological Vector Spaces
Now that we have established some primitive algebraic properties of
collections of sets, a natural question arises. If we are to do
analysis on setvalued functions, how do we take limits of sets? Well,
loosely speaking, we might say a sequence of sets (A ) converges to a
set A if the elements of the sequence are arbitrarily close to A. But
this poses another problem: how do we define "closeness" for sets?
Closeness for sets in a metric space can be defined in terms of the
Hausdorff metric (see section 3). For sequences of points in a general
topological vector space, J. von Neumann [18] introduced a notion of
convergence where closeness was defined in terms of a system of
neighborhoods of the origin. C.E. Rickart [20] later generalized this
notion to include convergence of sets in a locally convex topological
vector space, and it is Rickart's generalization that we present here.
1. Definition ref. 21 (p. 7). We say a vector space X is a
topological vector space if X has a topology T such that every point of
X is a closed set, and vector addition and scalar multiplication are
continuous with respect to T.
A collection A of neighborhoods of zero is said to be a local base
at zero if every neighborhood of zero contains a member of A.
We say a topological vector space is locally convex if there is a
local base at zero whose elements are convex.
A set V X is said to be balanced if aV V for any a e 0,
where laI < 1 and 4 is the scalar field associated with the vector space
X.
2. Remark. By Theorem 1.14 and its Corollary in ref. 21 (p. 1112),
every locally convex topological vector space X has a balanced convex
local base A at zero. We can use A to define a convergence of sets in X
that extends the usual notion of net convergence.
3. Definition. Let X be a locally convex topological vector space with
balanced convex local base A, and let A, B X. We say A and B are
equal within V E A ref. 20 (p. 500) if
A B + V and B S A + V.
4. Definition. Let (I, >) be a directed set, let (Ai) be a net of
iEI
subsets of a locally convex topological vector space X with balanced
convex local base A, and let A S X. We say (A.) converges to A, in
iEl
symbols
lim. A. = A,
1 1
if for any V E A, there is an i E I such that i > i implies A. and A
are equal within V
are equal within V.
11.3. Convergence in Metric Spaces
If two sets are in a normed linear space X, we may use the
definitions of the previous section to obtain a measure of the distance
between the sets. Here the balanced convex local base is the collection
of all neighborhoods
S (0) = {x 6 X: x < E,
and we say A and B are equal with E if
A B + S (0) and B S A + S (0).
This equality within e is also a special case of the Hausdorff
metric, which is a distance function on the nonempty bounded subsets of
an arbitrary metric space. It is this metric that we consider in this
section.
1. Remark. Throughout this section S will denote a set with a metric
d, and we say a set A S S is bounded if for any s S there is an
MS < such that d(a,S) < MS for all a E A; B will denote the
collection of all nonempty bounded subsets of S.
2. Definition ref. 13 (p. 167). Let A eB and B EB, and let
6(A,b) = inf d(a,b),
C(A, B) = sup 6(A, b),
where the infimum is taken over all a E A and the supremum is taken over
all b E B. Then we define the Hausdorff metric H by the formula
H(A,B) = max {C(A,B), C (B,A)}.
3. Note. H defines a metric on B if and only if we identify all sets A
and B which satisfy H(A, B) = 0. In a topological vector space X whose
topology is induced by a metric d, Lemma 4 along with Theorem 1.13(a)
ref. 21 (p. 11) will show that H(A,B) = 0 is equivalent to A = B. So we
say A is (closure) equivalent to B if A = B and H is a metric on the
collection of (closure) equivalence classes.
The relationship between equivalence within e and the Hausdorff
metric is clarified by the following Lemma:
4. Lemma. Let A B and B E B, then
H(A,B) = inf {r E R:r > 0, S (A) 2 B and Sr(B) 2 A}.
Proof. Let a = H(A,B), then for an arbitrary b e B and a E A,
a > 6(A,b) and a > 6(B,a). Thus Sa+E(A) 2 B and Sa+E(B) 3 A for
any e > 0, which implies that
H(A,B) > inf Ir E R:r > 0,S (A) B and S (B) 2 A}.
r r
Let e > 0 be given. There is an r > 0 with
Sr (A) 2 B, Sr (B) 2 A,
E E
and
r E inf {r E R:r > 0,S (A) 2 B,S (B) 2 A} + E.
Now S (A) 2 B implies (A,B) < r and S (B) 2 A implies C (B,A) < r
r Er
Therefore
H(A, B) < r
Since E was chosen arbitrarily,
H(A,B) < inf {r e R:r > 0,Sr(A) 2 B,Sr(B) 2 A}. O
Two useful concepts in defining the limit of a sequence of sets are
the concepts of upper and lower limits of the sequence.
5. Definition ref. 13 (p. 168). Let (A ) be a sequence of subsets
n=l
of S. Define A to be the set of all x in S such that there is a
sequence (a ) with an A for n = 1,2,. and a + x as n + m.
n=l
Define A to be the set of all x in S such that there is sequence of
positive integers (nk) and a sequence (a ) with a A for
k=l k=l k k
k = 1,2,. ., so that a + x as k + .
nk
6. Lemma. Let (A ) n= be a sequence from B that is Cauchy in H. Then
A = A.
Proof.
x E A.
Clearly A A. Also, if A = 0, it is clear that A A. So let
Then there is a sequence (nk)k=1 and a sequence (xk)k=1 with
xk EA
for k = 1,2,. ,
xk + x as k + m.
There is an mi from the sequence (nk)k=1 so that if n > mI and m > mI we
have
1
H(A,A) < .
If ml,...,mj1 have been chosen, we may choose mj from the
sequence (nk)k=l so that m. > mj. and if n > m., m > m. then we find
1
H(A ,A ) < 
n 2
Continuing in this fashion we obtain a sequence (mj.).j and a
corresponding subsequence (xm )j=l from the sequence (xk)k=l, so that
j
x
m.
J
as j m
and
1
H(A ,A ) <
n) m 2
whenever n,m > m..
J
22
Now we construct a sequence (bn)n= from the sequence (xm )j=, as
follows: if n < mi, we choose any z in An and set bn = z;
if mj < n < mj+l, choose z E A so that
1
d(x ,z) < ,
j 2j
and set b = z; and if n = mfor some j, set b = x Next we will
n n m.
oD
show that (bn)n= converges to x.
Given e > 0 there is a positive integer j so that
1 E
2j 4
Then for any n > m there is an mk > m so that
J J
mk < n < mk+l,
and there is an m > m so that
p k
d(x ,x) <
P
This implies
pl
d(b,x) <( d(b ,b ) + I d(b ,b
n n mk i=k mi mi+l
) + d(b ,x)
P
k i=k 21
2 i=k 2
Thus b + x as n + m, and this implies that x E A. Therefore
n
A A U
The statement of the next theorem, concerning the completeness of
the Hausdorff metric, does appear in the literature (see, for example,
Brooks ref. 5 (p. 312)). However, the proof of this theorem does not
appear, and so for the sake of completeness we give an original proof
here.
7. Theorem. Let (S,d) be a complete metric space. Then (8,H) is a
complete metric space.
Proof. Let (A )n be a Cauchy sequence of sets from B. We will show
n n=1
A + A as n co.
n 
Let E > 0 be given. Then there exists an integer N so that if
n,m > N we have
E
(1) H(An,Am <
Also, for any x in A, there is a sequence (x n) with x E A for
n = 1,2,..., and x x as n + . Thus there is an m > N so that
n S
d(xm,x) < .
For this xm, (1) implies that for any n > N there is an a e An so that
d(xm,a) < 
mI2
Hence for any n > NE, there is an a e A so that
Henc fo anyn >NEn
d(x,a) ( d(x,xm) + d(xm,a) ( E.
Therefore we find that for any n ) N ,
X E SE(An),
and since x was chosen arbitrarily in A,
A S (A ).
E n
There is an integer N1 so that for any integers n,m > N we have
H(An A ) < 1
n) m 2
Suppose N1,. . ,Nk1 have
Nk > Nk so that if n > Nk
k ki k
been chosen. Then we choose an integer
and m > Nk we have
H(A ,A ) < .I
nn k
2
Continuing this process by induction we obtain an increasing sequence of
integers (Nk)k=1 with the property that
H(AAm) <
2
whenever n,m > Nk.
Let E > 0 be given and choose a positive integer k0 so that
1
k 4
2
Let n be arbitrary with n > Nk0, and let z E An. Then there is a
k1 ) k0 so that
and there is a b with the property
1 k
k1
d(z,bk ) < 2
Now for each m > k1 we may choose, via induction, an element bm AN so
m
that
1
d(b ,b < 1
mlb m ) m1
In this way we obtain a sequence (bm)m=k, where bm Am and
1 m
m
d(bm,b ) < 2m
for m = 1,2, ..
Let a > 0 and let M be a positive integer, chosen sufficiently
large so that 2 < a/2. If m and n are integers satisfying n > m > M,
then
nI
d(b ) < d(bi,bi+)
i=m
nI
<
i =m
1 1
i m1
2 2
Hence the sequence (b )i=k is Cauchy. Since S is complete, there is an
x E S so that bi + x as i + . This implies x E A, and by Lemma 6,
x E A. Also, we see that there is a p > k1 so that
d(b px) < 4
Thus
p1
d(z,x) < d(z,bkl) +
1 i=k
d(bi,bi+l) + d(b ,x)
1 1 O
1 i =k 2
Therefore, for any n > Nk,
k0
A ES (A).
n E
Hence, for a given E > 0, there is an N = max {N ,NO} so that for
any n > N,
A S (A ) and A c S (A),
 n n 
that is,
H(A ,A) < e.
n 
This implies A B and
lim H(An,A) = 0. D
n+w
So the completeness of (S,d) is sufficient to guarantee the
completeness of (B,H) and its is easy to see that this condition is
necessary. For example, if (xn)n=1 is a Cauchy sequence with no limit
in S, then ({x })nl is a Cauchy sequence with no limit in B. Therefore
n n=1
the Hausdorff metric is complete if and only if its is defined on a
complete metric space.
11.4. Summation of Sets
Now that we have defined the concept of limit for sequences of
sets, can we defined the sum of a series of sets? The natural approach
is to take the limit of the sequence of partial sums. For example, in a
metric space we might say
y A = A
n
n=l
if
N
lim H() A ,A) = 0.
N+ n=l
However, a more stringent condition is needed to define a countably
additive setvalued set function.
Let (A )=1l be a countable disjoint collection of subsets of a
n n=1
measure space (S,E,I), and let (Bm)=1 be any arbitrary rearrangement of
the A 's. Then
n
U A = U B ,
n m
n=l m=l
which implies
I (A ) = i( An)
n=l n=l
= ( U Bm)
m=l
= (B )"
m=l
Thus not only must the series E (An) converge, but any
rearrangement of this series must also converge to the same value.
1. Definition ref. 14 (p. 959). Let X be a normed vector space and
let Z x be a series whose terms are in X. We say that Z x is
n n
n n
unconditionally convergent if
lim x exists in X,
Et TE
where the limit is a MooreSmith limit (see ref. 4 or ref. 16), and t is
the generic symbol for a finite subset of the nonnegative integers.
We say the sequence (Xn)n= is unconditionally summable if the
corresponding series E x is unconditionally convergent.
n
Now let X be a locally convex topological vector space with
balanced convex local base A, and let (B )n=l be a sequence of subsets
of X. We say (B )C. is unconditionally summable to B ref. 20 (p.50) if
n n=1
for every V s A, there is a t such that Z B and B are equal within V
it
for any n 2 tv
In the case where (B )n is a sequence of subsets of a metric
n n=1
space S, we say (B )n=l is unconditionally summable to B if for any
n n=1
E > 0, there is a it so that
H(B,E Bn) < e
whenever i : n t
2. Note. When X is a topological vector space whose topology is
generated by a metric d, then by Lemma 3.4, the two notions of
unconditional summability of sets in Definition 1 coincide for sequences
of bounded sets. Also, these definitions of unconditional summability
of sets extend the definition for sequences of points.
Another type of unconditional convergence of sets has been
introduced by Birkhoff ref. 3 (p. 362). Birkhoff uses the natural
approach of defining unconditional convergence of summablee selections"
from the terms of the set series.
3. Definition ref. 3 (p. 362). Let (A )n= be a sequence of subsets
of a Banach space X. We say (A )n=l is unconditionally selection
summable to a set A in X if any series E b (with b e B for
n n n
n
n = 1,2,...) is unconditionally convergent, and A is the locus of all
such sums.
This definition, apart from being a natural extension of
unconditional summability for sequences of points, seems easier to work
with than Rickart's Definition. Thus many of the more recent works
(e.g. [1,10,11,22]) employ definitions very similar to that of
Birkhoff. The next theorem shows Rickart's definition is at least as
general as Birkhoff's.
4. Theorem. (B ) n= is a sequence of subsets of a Banach space X and
 to B n n=1
if (Bn)n=1 is unconditionally selection summable to B, then (B )n=
is unconditionally summable to B.
Proof. Let (B )n=I be a sequence of subsets of a Banach space X such
n n=1
that (B )n= is unconditionally selection summable to B. Then given
n n=1
S> O, there is an N such that N < k < ... < k implies
c E 1 r
(1) D B k 2"
i=l 1
Define E = 1{,2,...,NE}. Then for any nt 2 e and any collection
{b } with b E B n there is a sequence (b ) = such that each element
n nEnE n n' n n=
of {b } appears as a term of (b ) and there is a b e B such that
n nEn n n=l
lim b = b.
n
Thus there is a n 2 n such that
E
(2) Ib I bn <
t'
31
So (1) and (2) now imply that for any n 2 i there is a b in B so that
b bn Ib b b b + l bn I bn
< + = E.
2 2
Hence for any n 2 n '
gT'
B E B + S (0).
n
Now given any b e B, there is a sequence (bn)n=l, with bn E Bn for
n = 1,2,..., such that
lim 1 b =b.
Thus there is a n 2 n (where n is defined as above) such that
C C.
Ib b <
nI
and consequently (1) and (4) imply for any n 2 E ,
Ib bn < .
Hence for any
Hence for any it 2 :: ,
B B + S (0).
n
%t
Therefore, (3), (5), and Lemma 3.4 imply
H(B,y Bn) (
n
it
whenever ni 2 TE which shows that (Bn)n=1 is unconditionally summable to
B. D
5. Note. The converse of Theorem 4 is not true in general. As an
example let X = R2 and define
B = {(x,y) E R21y = } for n = 1,2,...
n n
Then if i is any finite subset of the nonnegative integers containing at
least two elements we find
L3 n
B = R2.
n
Hence (B ) n_ is unconditionally summable to R but the sequence
((n,1))n1 certainly is not unconditionally summable to any element of
2 CO
R. So (B ) is not unconditionally selection summable. However,
n n=l
when the sequence (A ) consists of bounded sets, the two definitions are
equivalent.
6. Theorem. Suppose (An)n=l is a sequence of bounded subsets of a
Banach space X, and suppose (A ) is unconditionally summable to A, then
(An) is unconditionally selection summable to A.
Proof. By Note II.3.3 we may assume that the A 's and A are closed.
Let E > 0 be given. Since (A ) is unconditionally summable to A,
there is a it such that n D E implies
H(A, A A ) < E.
n
This implies that if (B )n= is an arbitrary rearrangement of (An)n=1
there is a natural number N so that m > N results in
m
H(A, X B ) < E.
n=l
The proof of Theorem II.3.7 shows that A is the set of all a in X with
I b = a
n
n=l
for some sequence (bn) with bn in Bn for n = 1,2, ..
Given (bn) with b E B there
n n n
exists a e A such that
m
exists N such that m > N implies there
0 E
Sb a < .
n m 3
n=l
In addition, there is an M so that if M < m < k then
E S
k
H(O, I Bn) < ,
n=m
which in turn implies
k
I \ b < <
n 3
n=m
So
k m k
ak aml 1 b ak + 1I b am + 11 bn E,
n=l n=l n=m
when m and k are chosen so that N v M < m < k. So the sequence (an)
is Cauchy; hence a + a for some a in A.
n
Hence for any E > 0 there is an L so that n > L implies
n 2
Thus
m m
a bn < a a + am bn < + < ,
n m n=l
n=l n=l
if we chose m > L v N Whence
E C
(1) b = a.
n
n=l
The above argument shows that any given rearrangement converges,
thus the sum in (1) is unconditionally convergent (see A of ref. 14).
Hence (An) is also unconditionally selection summable.
The next theorem characterizes infinite sums in the special case
where the summands are bounded closed intervals.
7. Theorem. Let (an)n= and (bn)n=l be two sequences of real numbers
with a < b for n = 1,2,..., and with
n n
X a = A, X b = B,
n n
n=l n=l
where A and B are real. Then
cD
X [a ,b ] = [A,B].
n=l
Moreover, if the sequences (a ) and (b ) n= are unconditionally
n n=1 n n=1
summable, then so is ([an,bn])n=l.
Proof. Given an E > 0, there is a natural number N so that
if k > N, then
k k
(1) an Al < E and \X b BI < .
n=l n=l
Applying Theorem 1.3 k 1 times, we find
k
(2) y [a ,bn] =
n=l
So (1) and (2) imply
k k
[I a i b ].
n=l n=l
k
I [a ,bn] c [A,B] + (E,E),
n=l
and
k
[A,B] i [an,b] + (bE,).
n I
n=1
Thus for any E > 0, there is an N such that if k > N, we have
k
H( [an,bn ],[A,B]) < E,
n=l
and we write
X [a ,b ] = [A,B].
n ,n n
n=l
Now let us suppose that (an)n= and (bn) n= are unconditionally
summable to A and B, respectively. Then given an E > 0, there is a
finite set of natural numbers n so that if n 2 'n we find
11 a Al < e and I b BI < E.
it iT
Hence, as above,
H(X [anbJ,[A,B]) = H([a a, bn],[A,B])< E,
whenever n 2 i This implies the sequence ([a ,b ])nl is
unconditionally summable to [A,B]. O
III. THE AINTEGRAL
The setvalued Aintegral will be central in our extension of the
YosidaHewitt decomposition. In section 1 we give the definition and
important properties of the integral. Of particular interest is the
result which shows that the integral is always countably additive. In
sections 2 and 3 we give sufficient conditions for the existence of the
Aintegral of a finitely additive set function.
III.1. Definitions and Basic Properties
In this section we review the definition and basic properties of
the Aintegral of a setvalued set function whose range lies in a
locally convex topological vector space. This integral was first
introduced by Rickart [20] as a generalization of work done by R.S.
Phillips and A. Kolmogoroff. In Rickart's paper, the results are stated
for a function defined on a measure space. Here we state his results
for a set function defined on an algebra of subsets. The same proofs go
through unaltered. These slightly more general results will be useful
later in generalizing the YosidaHewitt decomposition. All definitions
(except for Definition 8) and all theorems in section 1, excepting the
modification mentioned above, are due to Rickart [20].
1. Definition. Let S be a set and let E be an algebra of subsets of
S. By a subdivision A of S we mean an at most countable collection of
pairwise disjoint sets from E whose union is all of S. In what follows,
A (with or without subscripts) will always denote a subdivision of S.
Let Al A2 be subdivisions of the same set S, then we say A1 is
1
finer than A2, in symbols A > A2, if for every Ei E A1 there is a
2 1 2
E e A such that E S E The product of two subdivisions is a
ni 2 i ni
subdivision defined by
1 2 1 2
A E = E nE.:E e A1,E e A2j.
Let A0 = {E0} be a subdivision and (Ak) an arbitrary sequence of
k k k=1
0
subdivisions. Then the subdivision A which coincides with Ak on Ek for
all k is called the sum of (Ak) over AO, i.e.
0 E k 0 k
A = E nE:E A0, E Ak}.'
0~ E Ei K
In the remainder of this section we assume S is a set, E is an
algebra of subsets of S, X is a locally convex topological vector space,
A is a balanced convex local base for X, and i: + P(X) is a set
valued set function. We do not assume i is additive.
2. Definition. Given V e A and a decomposition A = {Ek} we use the
symbol J(p,A,A) to represent the sequence (i(A n Ek)) We say
k=l
J(i,A,A) is unconditionally summable to a set B with respect to V, if
for any Al > A there is a T1 such that n 2 Di implies B and E p(A n Ek)
are equal within V.
3. Definition. We say i is Aintegrable over a set A in E if there is
a set I(i,A) in X such that for any V c A, there is a subdivision A so
that A > A implies J(i,AA) is unconditionally summable to I(p,A) with
respect to V. The closure of the set I(p,A) is called the Aintegral of
i over A and is denoted
I(, A)c = f p(da).
A
If I(i,A) consists of a single element, then i is said to be
SAintegrable over A.
The Aintegral, when it exists, shares many of the desirable
properties of the more familiar Bochner integral.
4. Theorem. If i is Aintegrable over A, then the integral is unique.
5. Theorem. If 4 and Y are Aintegrable
then ai and p. + T are Aintegrable over A
fap(do) = a fp(do), Sfi(do) + T(do)
A A A
Although p may not even be additive,
the Aintegral of p must be additive.
over A, and a is a scalar,
and
= [f(do) + fY(da)]cl.
A A
the next theorem shows that
6. Theorem. If i is Aintegrable on both A and B, where A n B = 0,
then i is Aintegrable on A u B and
f A(da) = [ftl(d) + fIi(do)]
AUB A B cl
The Aintegral also has the unusual property that, no matter what
the additivity properties of i, the integral is always countably
additive. This stands in contrast to the integrals presented by Dunford
and Schwartz [12] and by Brooks [5] for finitely additive set
functions. With these later integrals, we expect finite additivity when
integrating a finitely additive set function.
7. Theorem. The Aintegral is a countably additive set function in the
sense that if p is Aintegrable over Ak for k = 0,1,2,..., where
A0= U Ak and the Ak's are pairwise disjoint for k> 1, then the
k=l
sequence
(f p(do))
Ak k=l
is unconditionally summable to f (do).
A0
Proof. There is no loss in generality in taking A0 = S. Given an
arbitrary but fixed V E A, since the integral exists for each Ak, there
k k
exist A such that if A > A then J(4,Ak,A) is unconditionally summable
v v
o k1
to f(do) with respect to 2k V for k = 0,1,2,. Let A be the
Ak
k 0
sum of (A ) over the subdivision {Ak, 1 and set A = A A On the
v k=l 0
set Ak, the subdivision AO is finer that A for any k > 0. Therefore
J(hp,Ak,A0) is unconditionally summable to f1i(do) with respect to
Ak
k1r
2 k. If A = {B}, then there is a nO such that n > nO implies
2 V. If
Sp(Bi) and f i(da)
it AO
are equal within V/2.
Define
n = max In:A n
v n
U B 4
n 0
Then for an arbitrary fixed n > n there exists a xn such that
v n
in 2 i E n, 1 > n imply A n B
n 0' n'1
= 0, and
1(Ak n Bi)
n
fp(do) X(Ak
Ak iEin
k n
C Jf (do)
Ak
+ 2k1,
+ 2 V,
n Bi) + 2k1
n Bi) + 2 V,
for k = 0,1,2,. ,n.
Since, for any i and k, Ak n Bi
K,
S(Ak
n
= or Ak n Bi = Bi, we have
n
Jp(Bi) E
n
fp(do) + V/2,
Ak
f p(do) c y(Bi) + V/2.
Ak Tn
Now since nt > ni we have
n 0
J i(do) E f i(dc) + V
Ak A0
thus
n
f(da) j(do) + V
A0 Ak
where n > n is arbitrary. This argument is independent of the order of
V
the Ak's, hence the result follows from Theorem 2.3 in reference 20. O
Borrowing a concept from Pettis [19] we can define a "weak" SA
integral:
8. Definition. Let X be a topological vector space with dual X Then
we say i is weakly SAintegrable over A in Z if there is an element
*
W(p, A) in X with the property that for any x E X x i is SA
integrable over A and
x W(i,A) = fx* i(do).
A
III.2. Existence of the SAIntegral
The definition of the Aintegral involves an assumption about the
existence of the integral in the space X. In this section we address
the question of when the integral exists and we give existence theorems
for the SAintegral of a set function 4 whose range lies in the real or
complex numbers. Throughout this section, as in the previous section,
we assume S is a set, E is an algebra of subsets of S, and i is a set
function defined on Z.
1. Theorem. Let i be a bounded, nonnegative, finitely additive set
function defined on E. Then for any E E i is SAintegrable over E.
Proof. Let A {A} and A {A } be two subdivisions of S
k
with A > A and UA = Ak for any k. Then the finite additivity of 1
implies that for any k,
Ei(E n Ai) < i(E n Ak) < .
i
Thus if A > A,
E i(E n Ak) = E 4i(E n Ak) < E (E n A ).
Ak k i AkA k
i
Now define
T = {Z p(E n A.)IA is a subdivision of S}.
A eA 1
Aied
Since pi is nonnegative, T is bounded below by zero. Set I = inf T.
Then for any e > 0 there is a subdivision A such that
I < E i(E n A.) < I + E/2.
A
So for any A > A ,
I < E (E n A ) < I + e/2.
A
Since i is nonnegative, the sum
E~(E n A.)
A
is unconditionally convergent (see [7]); hence there exists a nA such
that t > A implies
E(E n A ) EZ(E n A ) < E/2.
A t
Thus
I E < Ei(E n Ai) < I + E,
It
which implies that p is SAintegrable over E to I. O
2. Theorem. Let i be a realvalued finitely additive set function with
finite total variation defined on E. Then i is SAintegrable over any
set E in E.
Proof. Let L+ and p_ be the positive and negative parts of i, as in
reference 24, Theorem 1.12. Since i has finite total variation,
0 < +(S) < m, and
0 < x_(S) < .
Thus both i+ and i_ satisfy the hypotheses of Theorem 1; hence for any
E c E, i+ and p_ are SAintegrable over E and thus + +(1)t_ is SA
integrable over E. But i = p + (1)+_; hence i is SAintegrable over
E. O
3. Theorem. Let i be a complexvalued finitely additive set function
with finite total variation defined on E. Then i is SAintegrable over
any E c E, and if t = il + i 2, where ul and '2 are realvalued, then
f (do) = f p(do) + if p2(do). O
E E E
Proof. Since p has finite total variation, both il and +2 are
Svintegrable over any E in E by Theorem 2. Thus given any E in Z,
S= + ip2 is SAintegrable over E by Theorem 1.5. So
f l(do) = f (ti + i2)(do) = J ti(da) + i f 12(do).
E E E E
The next theorem shows that in certain cases the SAintegral may be
represented as a double MooreSmith limit. In this theorem X is a
** **
normed linear space with second dual X .We regard X X
4. Theorem. Let i:Z + X be finitely additive and sbounded. If
is SAintegrable over E E E, then
f p(do) = lim lim E i(E n E ).
E A TL n
E x
Note. In Theorem 4, the conclusion gives us
f p(do) E X,
E
even though it may happen that for a fixed A,
lim Ei(E n
nr
Proof. Considering i as having range
boundedness of i implies
in the Banach space X the s
in the Banach space X the s
SE(A) = lim Ei(E n E )
E TI n
it
exists in X whenever A = (E ) is a disjoint
[9]).
The SAintegrability of p over E implies
such that for any e > 0 there is a A so that
Swith the property
q with the property
sequence of sets in E (see
there is an I(,E) in X
A > A implies there is a
E
I(p,E) E4(E n EA) < e/2
n
whenever nr 2 iA .
i !
Also, for the same A, there is a tA so that if i 2 nt, we have
ISE(A) Ei(E n EA) < e/2.
Thus for any E > 0 there is a A so that A > A and n. 2 A u ~A
yields
II(,,E) SE(A)I < SE(A) Ei(E n EA)
+ II(, E) ZEI(E n E A) < e/2 + E/2 = E.
it
E ) E X
**\
\ X.
Hence
I(,E) lim S(A) li m S ) imA limx Eu(E n EA). O
It is doubtful that Theorem 4 can be generalized to hold for a
general SAintegrable set function a. The problem lies in line (1) of
the proof; Rickart's definition of integrability does not require this
limit to exist. For an informative discussion of these ideas see
reference 15.
111.3. Existence of the AIntegral
We start this section wth an existence theorem for the Aintegral
of a countably additive set function defined on a aalgebra of sets.
This theorem is the only existence theorem for the Aintegral found in
Rickart's paper.
1. Theorem. Let (S,E) be a measurable space, X a locally convex
topological vector space, A a balanced convex local base for X, and
i:E: + P(X) a countably additive set function. Then p is Aintegrable
over every E in E and
p(E) = fJ (do).
E
In the remainder of this section we expand Rickart's existence
theorem by providing existence theorems for a finitely additive set
function p over an algebra E of subsets of some arbitrary set S. We
first consider the case where the range of p is a collection of bounded
closed intervals on the real line.
2. Lemma. Let i be a bounded, closed intervalvalued set function
defined for E in E
p(E) = [a(E),P(E)]
Then
(1) p is finitely additive if and only if a and p are
finitely additive;
(2) i is countably additive if and only if a and P are
countably additive.
Proof. First we prove (2): Let (E )n= be a sequence of pairwise
disjoint sets from E such that U n= E = E e E. If i is countable
additive, then
additive, then
[a(E),3(E)] = E(E) = E p(En) = E
n=l n=l
This implies that for any e > 0 there is an N such that if k > N then
k
H([a(E),P(E)], E [a(En), P(En)]) < E.
n=l
So Theorem 11.1.3 implies that for any k > N,
k k
H([a(E),(E)], [E a(E ),E P(En)]) < E.
n=l n=l
[a(En), P(En)].
Thus we find
cO 00
[a(E),p(E)] = [E a(E ),E P(E )]
n=l n=l
which implies
a(E) = E a(E ) and P(E) = E P(E ).
n=l n=l
Hence a and P are countably additive.
Conversely, let a and P be countably additive. Then
W CO
a(E) = [a(E),P(E)] = [E a(En),Z a(E )],
n=l n=l
so Theorem 11.4.7 implies
(E) = Z [a(En), i( n).
n=l n=l
Hence i is countable additive.
By taking Ek = 0 for all k greater than some fixed N, the above
proof implies (1). D
3. Definition ref. 5 (p. 313). The total variation of a setvalued
set function i over a set E in E is given by
v(,E) = sup H(i(En),{O}),
n nn
4. Theorem. Let p be a closed intervalvalued finitely additive set
function defined on E with v(p,S) = M < . Then i is Aintegrable over
every E E.
If u(E) = [a(E),p(E)] for any E in E, then
f u(da) = [I(a,E),I(P,E)].
E
Proof. Define for each E in E
a(E) = min {x E R:x E (E)} and
P(E) = max {x e R:x E p(E)}.
Then for each E in E, p.(E) = [a(E),P(E)] and Lemma 2 shows that a and P
are finitely additive. Since v(i,S) = M, we find that H(p(E),{O}) < M
for any E E E, and so for any E E,
a(E) E [M,M] and p(E) E [M,M].
So Theorem 111.1.5 of ref. 12 implies a and P are of finite total
variation. Hence by Theorem 2.2, a and P are SAintegrable over any
E E.
Now let E e E be arbitrary, then for any e > 0 there is a partition
A such that for any A > A there is a "A such that 1 2 7A implies
(1) Ea(E n E.) I(a,E)[ < ,
and
(2) IEP(E n E ) I(,E) < E,
TL
where {E } = A. Also, we have from Theorem 11.1.3
EL(E n E ) = [Ea(E n E1), ZE(E n E)l,
so (1) and (2) imply
E(E n E ) S [I(a,E),I(p,E)] + (E,E), and
[I(a,E),I(P,E)] S E(E n E) + (E,E).
i9
Whence ui is Aintegrable over E, and
f i(do) = [I(a,E),I(p,E)]. O
E
Next we consider the case where the range of u is a collection of
finite dimensional "rectangles."
5. Definition. Let t be a setvalued set function defined on E with
values in Rn for some fixed n. We say is rectanglevalued if given
A E E,
p(A) = (II(A),I2(A),...,In(A)),
where I.(A) is a bounded closed interval in R for i = l,...,n.
6. Lemma. Let A1,BI,A2,B2,...,A nB be bounded subsets of the real
line. Then considering (A1, A ,...,A ) and (B B ,..., B ) as subsets
of Rn,
n
H((AI,A2 ...,An),(B1,B2 ...,Bn)) < E H(Ai,Bi).
i=1
Proof. Let E > 0. For convenience of notation let ai = H(Ai,B ) + E/n
n
and let a = E a Given an arbitrary
i=1
(x1,x2,...,Xn) in (A1,A2,...,An), then for each i = 1,2,...,n choose
y e B so that Ixi yil a This implies (xi yi)2< for
i = 1,2,...,n, and so
n 2 n 2 2
C (xi Yi) < ai a
i=1 i=1
Thus for each (x1,x2,...,xn) E (A1,A2,...,An) there is a
(y1,Y2"".'. n) e (B1,B2,...,Bn) with the property
I(xlx2,...,xn) (ylY2' ...yn l) < a.
Hence (A1,...,A ) (B1,...,Bn) + S (0). Similarly we find
(B1,...,Bn) (Al,...,An) + S (0). Lemma II.3.4 now implies
n
H((A1,...,An),(B1,...,Bn)) < a = E H(AI,Bi) + e.
i=1
Since E was chosen arbitrarily, the result now follows. O
7. Theorem. Let a: E + P(Rn) be a finitely additive rectanglevalued
set function with v(p,S) < m. If [p is defined on A E by
p(A) = (p(A),pi2(A),..., n(A)),
then i is finitely additive for i = 1,2,...,n. Furthermore, i is
Aintegrable over any A e E and
f p(do) = (f (l(da), f i2(da),..., f p (da)).
A A A A
Proof. Let A, B E Z such that A n B = 0. Then
(pl(A u B),..., pn(A U B)) = p(A U B)
= p(A) + p(B) = (p.(A),...,in(A)) + (pl(B),..., n(B))
= (G1(A) + 1(B),...,n(A) +n (B)).
So for i = l,...,n, i(A U B) = ii(A) + i(B).
Let {El,...,Ek be a finite disjoint sequence of sets from Z.
Then, for j = 1,...,n,
v(~,S) > =1i(E)H > i j (E)11.
1=1 i=l 1
Thus v(~,S) > v(pj,S), so ij has finite total variation for
j = l,...,n. Theorem 4 now implies ji is Aintegrable over any E E.
Let A E E and e > 0. Then for any j, there is a
subdivision A. such that A > A. implies there is a nA such that
J J
H(E p (Ei n A), f p.(dc)) < e/n,
iEi A
whenever n 2 A Let A be a product of Al ...,An (see Definition 1.1),
54
then A > A implies there is a 7 for each j, such that
H(E (Ei n A), J j(da)) < e/n,
ien A
whenever n 2 ." Thus A > A implies there is a nA
that i 2 tA implies
H(EZ (Ei n A), f 1i(do)) < e/n,
EXo A
for every j = 1,...,n, and so Lemma 6 implies
H(EZ (Ei n A), (f pl(do),...,
it A
n
= U=
f .n(da))
A
= H((E4I(Ei n A),...,En(Ei n A)),(f il(da),...,f n (da))
S7 A A
n
< E H(E (Ei n A),f p (do)) < e.
j=l ieS A
such
IV. RELATIONSHIPS BETWEEN THE AINTEGRAL
AND OTHER INTEGRALS
In the last chapter we defined the Aintegral and considered some
of the properties that follow from this definition. In particular, we
noted that the Aintegral of a set function (when the integral exists)
is always countably additive. This countable additivity is unusual; it
is independent of the additivity properties of the integrand. But the
Aintegral differs from the more familiar Bochner and Pettis integrals
in other ways as well; the Aintegral has an integrand consisting of
only one part, the function to be integrated, whereas the integrands of
Bochner and Pettis contain two parts, a function and a measure. So, the
question arises, how are these integrals related?
In section 1 we discuss the relationship between the Dunford
integral (which is a generalization of the Bochner integral) and the A
integral, and in section 2 we consider Artstein's generalization of the
Aumann integral.
IV.1. The Dunford Integral
In this section we assume that (S,E,p) is a measure space, X is a
Banach space, and f is an Xvalued function defined on S.
1. Definition. We define f(A), for any A S, to be the set of all
f(x) where x E A; in symbols
f(A) = {f(x):x E A}.
First we consider the case of f a simple iintegrable function.
2. Lemma. Let g be a simple pintegrable function on S, with i a
complexvalued measure. Define G on I by
G(E) = g(E)(E).
Then G is SAintegrable over any E E X.
Proof. Since g is simple we may write
g(s) = a lA (s) for a.e. s E S.
n=l n
Then G will be given by
m
G(E) = a nlA (E)i(E)
n= n
for any EE .
Now let E E Y and let A = {Bk} be any subdivision of S such that
{AI,...,A,(UAnf} < A.
Let E > 0 be given. The countable additivity of 4 implies there is a
i such that for any n 2 t ,
m m
1 a n I (E n Bk n A) a (E n A ) < .
n kkn n n
n=l ker n=l
Also, for any n 2 I,
m
JG(E n Bk) = anlA (E n Bk)(E n Bk)
i ken n=l n
m
= ) an (E n Bk n An),
n=l ken
Thus for any n 2 we have
m
)G(E n Bk) a an(E n A) < E.
t n=l
Hence G is SAintegrable on E and
m
(1) f G(da) = I an (E n A). O
E n=l
To utilize the ubiquitous process of taking limits of simple
functions in the case of the Aintegral, we need the following notion of
convergence:
3. Definition ref. 20 (p. 508). Let (S,E,i) be a positive measure
space and let (F ) be a sequence of setvalued set functions defined on
E. Then the sequence (F ) is said to converge approximately to F
relative to i if for every integer n and every e > 0, there is an A
and a subdivision A such that for each E,
n
lim [i(A ) = 0,
n E
and for A {Ek} > A we have that the sequences (Fn(A n Ek))k=1 and
(F(A n Ek))k=l are summably equal within E for every A S S \ An
E
4. Lemma. Let i be a complex measure defined on Z with
v(p,S) = M < m, and let (f ) be a sequence of Xvalued tmeasurable
simple functions on S converging to the function f ialmost
everywhere. For any n and any A E the Lebesque extension
of E, define
F (A) = f (A)u(A) and F(A) = f(A)u(A).
Then the sequence of set functions (Fn) converges approximately to F
relative to v(t).
Proof. Given n and e > 0, define
o(n,e) = {s e S: fn (s) f(s) > E/M).
By Lemma III.6.9 in ref. 12 (p. 147), we find o(n,e) E .
For a given n, f differs by a null function from a function of the
n
form
k
n
n
1. 1 (s),
1 "
i=l Ai
i
n n
for appropriate choices of a. and A.. So, for each n, define
1 1
n n
B. = A. \ a(n,E)
1 1
for i = 1,2,...,kn, and set
n n n
An = BB ,..' "'' k ,a(n,E)}.
E n
Then for any A = (a j > A ,E s 2 \ {o(n,e)}, t, and a c E n a we
have
I f (E n o.)X(E n o ) f(a )o(E n oaj)
< Ifn (E n o) f(a j) II(E n oj)
jEll
Si )i(E n a E.
Thus
H(X Fn(E n aj),)F(E n a )) < E;
that is, the sequences (F (E n a.)). and (F(E n aj))j are summably equal
within e for any E E \ {o(n,E)}.
Since (f ) converges in palmost everywhere to f we find, by a
n
corollary to Egoroff's theorem (Corollary III.6.13 in ref. 12 (p. 150)),
lim v(p,a(n,e)) = 0. [
n*M
Now we are in a position to give sufficient conditions under which
the Dunford integrability of a function f implies the Aintegrability of
the corresponding set function fix.
5. Theorem. Let 4 be a finite positive measure defined on E and let
f:S + X be iintegrable over any A E E. For any A e E define
F(A) = f(A)i(A).
Then F is SAintegrable over any E E E, and
fF(do) = ff dtp.
E E
Proof. Since f is uintegrable on S, there
measurable simple functions (f ) converging
such that
is a sequence of l
palmost everywhere to f
(1) lim ff di = ff di,
n+ E E
for any E E E. Let F (A) = f (A)p(A) for any n and any A E E. Then by
Lemma 2,
(2) fF (da) = ff dL,
E E
for any E E E. Hence for any E E E, (1) and (2) imply
(3) lim F (do) = ff du.
n+ E E
Since the iintegral of fn is absolutely continuous with respect
to v(i), (2) implies the SAintegral of Fn is absolutely continuous with
respect to v(i).
For any n, suppose
k
n
f (s) = E na 1 (s),
= A n
i
for a.e. s E S.
Then define A to be the partition
n
k c
n n n n n
{AA ,A2...,A A( ) ,
n i=1
(here we assume, without loss of generality, that the A ns are
disjoint). Then for any e > 0 and any A = {a } > An, there is a nE such
that i2 t implies
(4) 0 < ( (A ) E (Ai no < kM
jE n n
for all i = l,...,k where
M = max { ia : i = 1,...,k }.
n n
Hence for any E in E, (4) implies
0 < i(Ai n E) E (A n o n E) < M
jIrn nn
So, given e > 0 and A = {o } > A there is a ne such that nt 2 F
implies that for any E in E,
E F (E n o) fF (do)
jEn E
icitE
k k
n n
Sa ani(E no n An) (E n An)I
jEg ii= i (l
k
n
< 1l" I I i(E n An n j) a (E n An)
i=1 JI
k
n
< M ( ) = E.
n kM
i=1 n n
Hence F is uniformly SAintegrable. By Lemma 4, (F ) converges
n n
approximately to F. So by Theorem 9.5 in reference 20, F is SA
integrable over any E in E, and
lim fF (do) = fF (do).
n+ E E
Now (3) implies
fF (do) = ff dp..
E E
6. Corollary. Let i be a complexvalued measure defined on E, and let
f:S + X be a function which is 1integrable over any E in E. For any E
in E define
F(E) = f(E)l(E).
Then F is SAintegrable over any E in E and
fF (do) = ff d..
E E
Proof. Write i = i l 2 + i(I3 14) where )i > 0 for i 1,...,4. By
Corollary III.4.5 in reference 12, is bounded and so ai is finite for
i = 1,...,4. Since 0 < i < v(4), we see that f is i integrable over
any E E for i = 1,...,4. For each i and each A in E, define
Fi(A) = f(A)Ii(A).
With this definition, Theorem 5 yields
fFi(do) = ff dpi,
E E
for any E in E and any i = 1,...,4. Now and Theorem III.1.5 imply F
is SAintegrable over any E e Z and
fF (do) = fF1 (do) fF2 (dc) + i(fF3 (da) JF4 (dc))
E E E E E
= ff d~l ff dv2 + i(ff di3 ff d 4)
E E E E
= ff dp.
E
IV.2. The Aumann Integral
Recently several authors (e.g. [1,2,10,11,19,22]) have considered
the problem of integrating a setvalued function with respect to a
singlevalued measure. The integration technique first defined by R.J.
Aumann [2] appears to be the most popular method for such functions,
probably due to the simplicity of definition. The definition we give
here is a slight generalization of Aumann's definition, due to
Artstein. In this section we assume (S,E,) is a nonnegative measure
space and F is a function defined S with values in P(Rn)
1. Definition ref. 1 (p. 116). Suppose the set {s e S:F(s) = 0} is i
null. Let F be the collection of all functions f that are iintegrable
over S (in the sense of Lebesgue) such that f(s) e F(s) ualmost
everywhere. Then the Aumann integral of F over E c E is defined by
fF di = {ff di : f F }.
E E
To show a connection between Aumann's integral and the A integral,
we need the following definition.
2. Definition. Let A and B be sets and let F:A P(B) be a setvalued
function. Then for any E c A define
F(E) = u F(e).
eE
With this definition we can show that when the Aintegral
of Fp exists, it contains the Aumann integral of F with respect to ..
3. Theorem. Let F have the property F(s) ; 0 for any s E S. If we
define G on Z by
G(A) = F(A)p.(A)
and if G is Aintegrable over E c E, then
fF di fG(dc).
E E
Proof. If the Aintegral of G exists over E E then by definition,
I I
for every E > 0 there is a A e such that A > A E implies there is a nA
so that if i 2 xA then
(1) LG(E n E1) E JG(da) + S (0) or
ET E
(2) )F(E nE) Ai (E n E) C fG(dc) + SE(0).
it E
Let f be function which is integrable (in the sense of Lebesgue)
over S and f(s) e F(s). Then Theorem 1.5 implies that f is SA
integrable over E. By the definition of SAintegrability there is a
A such that A > A implies there is a A with the property
(3) ff(da)1(do) E ]f(E n E )p(E n E ) + S (0)
E it
for every ni 2 .
I WI
Let A = A A then (2) and (3) yield
E E E
ff d i = ff(da)ji(da)
E E
e Ef(E n Ei)(E n Ei) + S (0)
SEZF(E n EA )(E n E) + S (0)
it
E G(do) + S (0)
EE
for sufficiently large n. So, we find
(4) ff d E [JG (da)J = fG (da).
E E E
Since (4) holds for every Lebesgue integrable selection of F, the
definition of the Aumann integral yields
fF di S fG (do).
E E
V. APPLICATION OF THE AINTEGRAL TO DECOMPOSITION THEOREMS
In this chapter we demonstrate, for a finitely additive set
function L, a relationship between the Aintegral of p and the Yosida
Hewitt decomposition of p. This relationship allows us to extend this
decomposition to the case of finitely additive setvalued set functions.
In 1952, Kosaku Yosida and Edwin Hewitt [24] proved that a bounded
realvalued finitely additive set function defined on an algebra can be
written as the sum of a countably additive set function and a set
function which is "purely" finitely additive. Their proof relies
heavily on the lattice properties of the space of bounded finitely
additive realvalued set functions defined on an algebra E.
In 1969, James K. Brooks [6] gave a concise, elegant proof which
extended the YosidaHewitt decomposition to finitely additive set
functions of bounded semivariation whose values lie in a Banach space
X. Under these general conditions Brooks considers decomposing his set
functions in X the second dual of X. When X is reflexive, he shows
that the set function p (with the conditions stated above) can be
decomposed into a countably additive function with values in X and a set
X*
function T such that x Y is purely finitely additive for each x in X
the dual of X.
J.J. Uhl, Jr. in 1970, decomposed a finitely additive Banach valued
set function into a countably additive part and a weakly purely finitely
additive part (as above) both of which have values in X. He requires
the set function to be absolutely continuous with respect to some
nonnegative finitely additive set function. While Uhl's work is not as
general of Brooks', he does provide an interesting proof by projecting
the problem into the Stone space E of the algebra E.
We provide, in this chapter, an entirely new approach to this
problem. We show that the YosidaHewitt decomposition can be attained
in a constructive manner via the Aintegral. In section 1 we treat the
case of scalarvalued set functions, in section 2 vectorvalued set
functions, and in section 3 setvalued set functions.
V.1. Scalarvalued set functions
Throughout this section we assume S is a set and E is an algebra of
subsets of S. Unless stated otherwise we assume t is a finitely
additive realvalued set function defined on E.
1. Definition ref. 24 (p.48). For set functions p. and Y on E, we say
L Y> if and only if i(A) > Y(A) for every A in E. Let > 0, then we
say p is purely finitely additive if for any countably additive set
function Y on E such that 0 < T < p, we have = 0. When 4 is
realvalued, we say pi is purely finitely additive if both + and i_ are
purely finitely additive.
2. Example (1) Let S = (0,1] and let Z be the algebra consisting of
all finite unions of the form
(1) We are indebted to H. Gryzbowski for suggesting this example.
k
u (a ,b] where a ,bn E [0,1].
n=l
Define the set function i on Z as follows:
1
I(A) =
0
Let A and B be disjoint
then
if (2, +
otherwise.
sets from E.
E) A for some E > 0
If (2' + E)
A for some E > 0,
p(A u B) = 1 = (A) + i(B).
If neither A nor B contains a set of
the form (, 2 + E) then
p(A u B) = 0 = +(A) + i(B).
Hence i is finitely additive.
Now let Y be an arbitrary countably additive set function on E such
1
that 0 < p. Restricted to the interval (0, 2], we find = 0. For
each interval of the form I = ( + 1 + ], where n > 2, we have
n f + +1' 2 n
j(In ) = 0; hence (In) = 0.
n n
Furthermore
((' 11i = Y2(In) = O.
2n=2 n
Therefore T is identically zero on E, which implies that p. is purely
finitely additive.
The next theorem shows that the Aintegral is in fact the countably
additive part of the YosidaHewitt decomposition.
3. Theorem. Let pi be a nonnegative, finitely additive bounded set
function on E. Then for any A in E we can write
p(A) = f p(do) + x(A)
A
where the set function
f p(do)
(.)
is countably additive and x is purely finitely additive.
Proof. The Aintegral of p exists over each A in E by
and the countable additivity of this integral is given
11.1.7.
Suppose X is a nonnegative countably additive set
such that X < p. Then for any A in E, any subdivision
iT, we have
Theorem 11.2.1,
by Theorem
function on Z
A = {Ek}, and any
(1) I X(A n Ek) < i(A n Ek) < p(A).
By definition of the SAintegral, given c > 0, there is a
subdivision A so that for any A = {Ek} ) A there is it such that
k A
(2) 11 (A n Ek) I(,A)I < E,
it
whenever
For
t
the same A, there is a n' such that n 2 x implies
(3) X(A) E < kX(A n Ek).
Thus for any n 2 TA u ~', (1), (2), and (3) together imply
X(A) E < ) X(A n Ek)
< i(A n Ek)
< I(i.A) + c < i(A) + 2c.
Since E was chosen arbitrarily, we find
(4) k(A) < I(p,A) < i(A)
for any A in E.
Define x on Z by the equation
x(A) = i(A) I(L,A).
Suppose T is a countably additive set function defined on E with
0 < ? < x. Then 0 < W < a I(), which implies Y + I(u) < p.
Since T > 0, we have
I(L) < T +
However, + I(p), being the sum of two countably additive set
functions, is countably additive, so (4) implies
I(0) = Y + I(),
which in turn implies T = 0. Therefore x is purely finitely additive. O
As an illustration of the above theorem let us compute the SA
integral of the set function defined in Example 2.
4. Example. Let A c (0,1] such that , + e] A for some e > 0.
22
Consider the decomposition
A = (0, ],( + + ]: n > 2
S2 2 n + 1 2 n
If A > A then J(i,A,A) is the sequence consisting of all zeroes.
Hence J(i,A,A) is unconditionally summable to zero. The same happens if
A does not contain an interval of the form (, + E), thus i is SA
integrable over any A in Z and
J{i(do) = 0.
A
So, in this case at least, the conclusion of our theorem is valid; for
any A in E,
I(G) < 4.
4(A) = fi(do) + [(A).
A
The next theorem shows that we can relax the restriction of
nonnegativity.
5. Theorem. Let i be a realvalued with v(4,S) < m. Then for any A in
E we can write
4(A) = f[(do) + x(A),
A
where f4(da) is countably additive and x is purely finitely additive. O
Proof. The existence of the integeral is guaranteed by Theorem 11.2.2,
and its countably additivity follows, as above, from Theorem II.1.7.
Define for any E in E, x(E) = i(E) I(4,E). Let p = + i_ be
the Jordan decomposition of 4, as in Theorem 1.12 of ref. 24. Then by
Theorem 11.1.5,
ft(do) = f+(da) f _(do).
E E E
Hence
(1) x(E) = (4(E) fJ (do) i_(do))
E E
for any E in E. From (4) in the proof of Theorem 5, we find that
4 (E) f (do) and 4_(E) fl_(do)
E E
are nonnegative for any E in .and, again by theorem 5, the set
functions + J +(do) and i_ Ji_(do) are purely finitely additive.
Now (1), and Theorem 1.17 of ref. 24, imply that x is purely finitely
additive.
The following theorem, a generalization of Theorem 1.18 in ref. 24,
shows that given a purely finitely additive set function and a countably
additive set function, their masses are distributed on different parts
of the underlying space.
6. Theorem. Let (S,E) be a measurable space and let ui be a finitely
additive, realvalued set function on E with v((,S) < m. Then i is
purely finitely additive if and only if for any countably additive real
valued set function with v(Y,S) < , any A in E, and any E > 0, 6 > 0,
there is a T e E with T c A, v([,T) < E and Y(A T') e (6,6).
Proof. By Theorem 1.12 in ref. 24, p n [_ = 0, so Theorem 1.21 in [24]
implies that given E, 6 > 0, there is a set E c E and a set B E E such
that i+(E) < ,, _(E') < , +(B) = 0, and Y_(B') = 0. So given any
F in E, we have
S(E n F) < C, i_(E' n F) < ,
+(E
(0)
S+(B n F) = 0, and '_(B' n F) = 0.
Now we apply Theorem 1.18 of ref. 24 four times: given
(1) there is a T1 c E' n B n A such that + (T) < and
S_(T' n E' n B n A) < ;
1
(2) there is a T2 c E' n B' n A such that
S(T2) < and + (T' n E'
6
n B' n A) <
(3) there is a T3 E n B n A such that
1 (T3) < and T_(T' n E n
3 83
6
B n A) < ;
(4) there is a T4 n E n B' n A such that
4_(T ) < and Y+(T n E n B' n A) <
unesto tht4
In (1) through (4) it is
Let T = T1 u T2 u T3
understood that T. E .
u T Then
(T{ n E' n B n A) u (TB n E' n B' n A) u (TI n E n B n A)u
(T4 n E n B' n A)
= A n [(E'
n B n TI) u (E n B n Tj) u (E n B' n TA) u (E n B' n T')]
= A n [(E' n B n T') u
(E' n B' n T') u (E n B n T') u (E n B' n T')]
= A n [((E' n B) u (E' n B') u (E n B) u (E n B')] n T']
= A n S n T' = A n T'.
Now (0) through (4) imply
v(L,T) = L+(T) + u_(T)
4
i=1
4
< ( + ) =
i=1
Y(A n T') = I (AnT') Y_(A n T')
= +(T' n E' n B' n A) + Y (T' n E n B' n A)
+2 4
 _(T' n E' n B n A) Y_(T' n E n B n A) E ( 6,6).
1 3
Conversely, suppose T is an arbitrary nonnegative countably
additive set function with v(Y,S) < . By hypothesis, for any E, 6 > 0
and for any A in E, there is a set T in E such that T A and
v(i,T) < E, T(A T') < 6. But then v(i,T) < c implies p+(T) <
and i (T) < E. Thus Theorem 1.18 in ref. 24 implies the desired result. O
7. Definition. Let i be a finitely additive complexvalued set
function defined on E. We say i is purely finitely additive if both
R e and I m are purely finitely additive.
e m
8. Theorem. Let t be finitely additive and complexvalued with
v([,S) < m. then we can write
i(A) = Jf(do) + x(A), for A E,
A
where x is purely finitely additive.
Proof. The existence of the integral follows from Theorem 11.2.3, and,
as above, the integral is countably additive.
Let i = pl + il2, where i1 and p2 are real set functions. Now
define, for any A in E,
x(A) = I(A) fp(da)
A
= I(A) + i2 (A) f l(do) if 2(do)
A A
= I(A) fJ(do) + i([2(A) fJ2(do)).
A A
So Theorem 5 implies that the real and imaginary parts of x are purely
finitely additive. Thus x is purely finitely additive.
V.2. Vectorvalued Set Functions
Building upon the theorems of the previous section, we now extend
the YosidaHewitt decomposition to vectorvalued set functions.
Throughout this section S will represent an arbitrary set, E an algebra
of subsets of S, and i a finitely additive set function defined on E.
*
1. Theorem. Let X be a topological vector space with dal space X
Let i be a finitely additive Xvalued set function that is weakly SA
*
integrable over any E in E and v(x i,S) < for any x E X Then we
can write
u(E) = W(,E) + x(E)
for any E in E, where W(L) is weakly countably additive and x x is
*
purely finitely additive for any x e X
If X is a normed linear space this decomposition is unique.
Proof. Given E in E, let W(u,E) denote the weak SAintegral of i over
E. Then the countably additive of the SAintegral implies the weak
countably additivity of the set function W(4).
Now define for any E in E,
x(E) = a(E) W(p,E).
*
Then for an arbitrary x e X and any E E E,
*
x x(E) = x (E) x W(p,E)
=x (E) Jx *(da).
E
So by Theorem 1.8, x p is purely finitely additive.
Suppose X is a normed linear space, and suppose
S= W(pI) + x = x + x
c p
where x is weakly countably additive, x x is purely finitely additive
*
for any x c X and W(i) and x are as above. Then for any x X we
have
*
x W(G) x c = x p x x,
where the left side of this equation is countably additive, and the
*
right side is purely finitely additive. Hence for any x E X ,
*
x W(Oi) = x ),
*
x jp = x.
So by a Corollary of the HahnBanach Theorem (II.3.14 in ref. 12) we
find
W(G) =1,
c
and
tp =x,
that is, the decomposition is unique.
2. Corollary. Let [i satisfy the hypotheses of Theorem 1, where we now
take E to be a aalgebra and X to be a Banach space. Then we can write
i uniquely as
S= W(G) + x,
where W(g) is countably additive and x x is purely finitely additive for
any x X
any x EX .
Proof. By Theorem 1, L = W(L) + x uniquely, where for
*
any x c X x W(p) is weakly countably additive and x x is purely
finitely additive. By the Pettis theorem (IV.10.1 in ref. 12), W(i) is
countably additive. O
The following theorem generalizes the extensions of the Yosida
Hewitt decomposition given by Brooks [6] and Uhl [23], both of which
provide decompositions for Banachvalued functions. Uhl imposes the
more restrictive condition that the set function be sbounded (see ref.
8). Our conditions and subsequent decomposition are similar to those of
Brooks [6], although our theorem holds for functions whose values lie in
a complex normed linear space.
3. Theorem. Let X be a normed linear space with dual X and second
**
dual X Let u be a bounded, Xvalued, finitely additive set
function. Then for any E in E, we may uniquely write
u(E) = W(p,E) + x(E),
** ** *
where W(p,E) and x(E) are elements of X .For each x E X W(p)x
is countably additive and xx is purely finitely additive.
Proof. For any E e define W(p,E):X C by
W(I,E)x = fx p(do).
E
x* *
For any x EX Ix i(E)l < Ix p(E) < Ix j(E), so x t is
bounded. Now by Lemma III.1.5 in ref. 12, x p is of bounded variation,
and so by Theorem III.2.3, x i is SAintegrable over any set in E. Thus
*
given E E E and x E X W(p,E)x is well defined.
Given x y E X and a a scalar, then
*
W(P,E)(x + ay )
= fx*(dao)
E
= f(x + ay )p(do)
E
+ afy *(do)
E
W(,E)x + W(,E)y
= W(I1,E)x + aW(i,E)y ,
by Theorem 111.1.5. Hence for any
X .
given E in E, W(p,E) is linear on
*
Fix E E E. Then given x e X for any E > 0, there is a
subdivision A = {E } and a t such that
n E
ifx* (do) x* (E n E ) < .
E TE
E
This implies
IW(G,E)x* = Ifx%(do)I
E
< fxJ (do) ) x p(E n E)I + x1 x (E n En)
E n n
E i
(E n
< + [x [ i (E n E ) n + Ix *p(E).
Hence
IW(U,E) sup IW(i,E)x* < (E) < m,
**
and so for any given E E, W(p,E) E X
*
For a given x e X the countable additivity of I(x .) implies the
*
countable additivity of W()x .
Define x on E by the equation x(E) = i(E) W(i,E). Then for any
*
x EX,
*
xx = x I W(p)x
= x i fx (da),
*
and so by Theorem 1.8, xx is purely finitely additive.
To prove uniqueness, suppose i = uc + Lp is any decomposition
satisfying the conclusions of the theorem. Then, as in the proof of
Theorem 1, given E E ,
*
W(p,E)x = i(E)x ,
and
*
x(E)x = u (E)x ,
p
*
for any x X Hence given E c W(p.,E) and pc(E) define the same
functional on X as do x(E) and i (E).
4. Corollary. In addition to the hypotheses of Theorem 3, suppose X is
reflexive. Then we may write
u(E) = W(u,E) + x(E)
for any E e E, where W(p,E) and x(E) are in X. If we also assume E is
a oalgebra, then W(i) is countably additive.
Proof. By the Pettis theorem (IV.10.1 in ref. 12), the weak countably
additivity of W(p) given in Theorem 3 implies strong countably
additivity.
V.3. Setvalued Set Functions
In this section we consider a direction of generalization of the
YosidaHewitt decomposition that, to the author's knowledge, has never
been undertaken. As in the previous sections we will take E to be an
algebra of subsets of some set S. But in this section we will
consider L to be a finitely additive setvalued set function.
1. Definition. Let X be a normed linear space and let i be a P(X)
valued finitely additive set function defined on E with v(u,S) < .
Then we say t is purely finitely additive if v(i) is purely finitely
additive.
2. Theorem. Let p be a finitely additive closed intervalvalued set
function with v(p,S) < . Then p can be written as
li(E) = fj(da) + x(E)
E
for E E,
where fI(do) is countably additive and x is purely finitely additive.
Moreover, this decomposition is unique up to closed convex sets.
Proof. For any A e E, define
a(A) = min [(A),
P(A) = max i(A).
Then p(A) = [a(A),P(A)], and by Theorem III.3.4,
fJ(da) = [I(a,A),I(p,A)].
A
From (3) in the proof of Theorem III.3.4, given A E E and
any E > 0, there is a partition A = {E } and a finite set T so that
n
fp(da) P(A n E ) + ( I)
A i
This implies that
I(P,A) I(a,A) < P(u (A n E )) a(u (A n E )) + E
n n
< P(A) a(A) + e.
So by Corollary II.1.4, we can write, for any A E Z,
p(A) = JC(do) + x(A)
A
where
x(A) = [a(A) I(a,A),3(A) I(P,A)].
Define xl and x2 on E by
x1(A) = a(A) I(a,A),
x2(A) = P(A) I(P,A).
By Theorem 1.5, xl and x2 are purely finitely additive. Hence by
Theorem 1.17 in ref. 24, the set function f, defined on Z by
f(A) = v(xl,A) + v(x2,A),
is purely finitely additive. So we find, by Theorem 1.6, that for any
countably additive set function Y defined on E, for any E > 0,
6 > 0, and for any A E E, there is a subset T of A in E with f(T) < E
and '(A n T') E (6,6). Whence
v(x,T) = sup 11x(Tn )I
< sup (Ixl(T n) + Ix2(Tn)I)
< sup .[x1(T l)I + sup lIx2(T 2n)
= v(x1,T) + v(x2,T)
= f(T) < E,
where the supremums are taken over all finite disjoint partitions of
T. Theorem 1.6 thus implies v(x) is purely finitely additive. The
countable additivity of the integral follows from Theorem III.1.7.
To prove uniqueness, suppose u = c + p', where c is countably
additive, p is purely finitely additive, and
P
S[clIc2 = I'c2' p = pl'1p2
From i = I(i) + x and p = c + [p it follows that
(1)
(2)
I(a) + x = 1cl + pl
I(P) + x2 =c2 + p2*
Adding (1) and (2) and transposing yields
I(a) + I(p) cl c2 = pl + x x2
Since the left side of this equation is countably additive and the right
side is purely finitely additive, we find
I(a) + I(B) = 1cl + c2
(4) ppl + p2 I xl + x2
Subtracting (2) from (1) and transposing yields
I(a) I(P) + nc2 cl = pl
 p2 + x2 xl'
As above, we find that this implies
x1 2 = #pl
Now adding (3) and (5), subtracting (5) from (3), adding (4) and
(6), and subtracting (6) from (4) implies
I(a) = cl, I(P) = c2'
xl = pl' and x2 = p2"
This establishes the uniqueness of the decomposition. O
Finally, we give a decomposition for finitely additive set
functions whose values are rectangles in Rn (recall that a rectangle is
a cross product of closed intervals).
3. Theorem. Let i: P(R") be a finitely additive rectanglevalued
set function with v(p,S) < . Then for any E e E, we can write
I(a) I(P) = cl .c2
 p2"
p(E) = fi(do) + x(E),
E
where fi(do) is countably additive and x x is purely finitely additive
n
for any x X = R
Proof. Theorem 111.3.7 guarantees the existence of the Aintegral
of over any E E, and
f i(do)
E
U (f1.(do),
E
By Theorem 2, for j = l,...,n, there is a set function x : + P(R)
with the property
j (E) = Jij(do) + x.(E), for any E E.
E
Define x on E by x(E) = (xl(E),x2(E),...,xn(E)).
Then for E E,
i(E) = ([l(E),..., n(E))
= (1l(do) + x1(E),...,f (do) + Xn(E))
E E
= (nl(do),...,fn(do) + (xl(E),...,xn(E))
E E
= fJ(do) + x(E).
E
Throughout the remainder of the proof, let x be an arbitrary (but
fixed) element of X
' nn(o )
E
Let K be a compact connected subset of Rn. The continuity of x
implies that x K is also a compact connected set in R, and so x K is a
*
bounded closed interval in R. This implies that x ., x fp(da), and
x x are all bounded closed intervalvalued set functions.
By Theorem 7.1 in ref. 20, x p is Aintegrable with
x *fi(do) = Jx *(do),
E E
whenever E E E. So we now have
x p(E) = fx p(da) + x x(E).
Let (E.) be a finite disjoint sequence from E. Then
;lx*%(Ei)l < Ix*l HS(E i)
= [x*I 1t(E) 1
< Jx*J v( ,S) < ,
thus v(x p,S) < Ix v(p,S)
< m. Theorem 2 now implies
x *(E) = fx p(da) + y(E),
where y is the purely finitely additive set function defined by
y(E) = [min x p(E) min fx *(da), max x p(E) max fx *(do)].
As in Corollary 11.1.4, the bounded, closed interval ZE which solves the
equation
x* (E) = fx* (do) + ZE
E
for a given E in E is unique. Thus for E in Z,
x x(E) = y(E),
and so x x is purely finitely additive. D
BIBLIOGRAPHY
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BIOGRAPHICAL SKETCH
Michael Joseph Sousa was born on March 20, 1959 in Ft. Lauderdale,
Florida. He grew up in Ft. Lauderdale where he attended Piper High
School. After completing his secondary education, he entered the
University of Florida in Septemeber, 1977, as a Sociology major. He
worked as a research assistant under Dr. Charles Wood and nearly
completed his coursework in Sociology when he was drawn back to his true
love, Mathematics. After graduating with honors in 1981, he entered the
University of Florida's graduate school in Mathematics. Working under
the direction of Dr. James K. Brooks, he developed an interest in
Measure and Integration theory. His other interests include computer
architecture and fast algorithms for digit signal processing.
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 degre of Doctor of Philosophy.
Dr. James Kl Brooks, Chairman
Professor of Mathematics
I certify that I have read this
to acceptable standards of scholarly
scope and quality, as a dissertation
I certify that I have read this
to acceptable standards of scholarly
scope and quality, as a dissertation
I certify that I have read this
to acceptable standards of scholarly
scope and quality, as a dissertation
I certify that I have read this
to acceptable standards of scholarly
scope and quality, as a dissertation
study and that in my opinion it conforms
presentation and is fully adequate, in
for the degree of Doctor of Philosophy.
/ / I
Dr. Nicolae Dinculeanu
Professor of Mathematics
study and that in my opinion it conforms
presentation and is fully adequate, in
for the degree of Doctor of Phi osophy.
Dr. Louis Block
Associate Professor of Mathematics
study and that in my opinion it conforms
presentation and is fully adequate, in
for the degree of Doctor of Philosophy.
Dr. Jorge Martinez
Professor of Mathematics
study and that in my opinion it conforms
presentation and is fully adequate, in
for the degree of Doctor of Philosophy
Dr. William Dolbier
Professor of Chemistry
This dissertation was submitted to the Graduate Faculty of the Department
of Mathematics in the College of Liberal Arts and Sciences and to the Graduate
School, and was accepted as partial fulfillment of the requirements for the
degree of Doctor of Philosophy.
December 1985
Dean, Graduate School
