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 Title: Set-valued integrals
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 Material Information Title: Set-valued integrals Physical Description: Book Language: English Creator: Sousa, Michael Joseph, 1959- Copyright Date: 1985
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SET-VALUED 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

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.

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 A-INTEGRAL............................... ........... 37
III.1 Definitions and Basic Properties..........37

111.2 Existence of the SA-integral..............42
111.3 Existence of the A-integral...............47

IV. RELATIONSHIPS BETWEEN THE A-INTEGRAL
AND OTHER INTEGRALS ..............................55
IV.1 The Dunford Integral.......................55

IV.2 The Aumann Integral........................ 63

V. APPLICATION OF THE A-INTEGRAL TO
DECOMPOSITION THEOREMS........................... 67
V.1 Scalal-valued Set Functions................. 68
V.2 Vector-valued Set Functions.................77

V.3 Set-valued 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

SET-VALUED 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 A-integral, the

integrability of a finitely additive set function p is considered.

When i has finite total variation and is complex-valued, 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 complex-valued measure u, then the set-valued set

function flI is A-integrable, 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 A-integrable, then the A-integral contains the Aumann

integral. Finally, Yosida-Hewitt 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 set-valued functions and set-valued

integrals has enjoyed an increasing popularity. Set-valued functions

have been applied in areas as diverse as mathematical economics and

numerical analysis. Nobel laureate Gerard Debreu boosted interest in

set-valued 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 a-finite positive

measure space (A,E,p), where A is a set of agents, and E is the o-field

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

Radon-Nikodym 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 round-off

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 set-valued 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

finite-dimensional real vector space. When the measure is bounded and

nonatomic, it is convex-valued. He was also able to prove that when a

set-valued measure i with convex values is absolutely continuous with

respect to a finite nonnegative measure X, then i has a Radon-Nikodfm

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 a-ring

generated by R, and provides Vitali-Hahn-Saks 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 set-valued

set functions defined on an algebra with values in the power set of Rm

These recent studies do not imply that the theory of set-valued

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 so-called 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 set-valued function is also not new. In

1930, A. Kolmogoroff considered the integration of real set-valued

functions defined on a measurable space, but he required the integral to

be single-valued. 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 single-valued integrals.

C.E. Rickart [20] showed in 1942 that he was willing to consider a

set-valued 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 A-integrable 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 set-valued 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 n-space

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

Banach-valued function against a finitely additive set-valued 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

Banach-valued 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 p-measure, 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 real-valued function f is said to be weakly p.-

integrable if f is x p-integrable 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 A-integral and the Yosida-Hewitt

decomposition of a finitely additive set function. The Yosida-Hewitt

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 set-valued 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 set-valued 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 A-integral, all of which are original except for

Theorem 3.1, which is due to Rickart.

Chapter IV gives relationships between the A-integral and the

Dunford integral, and between the A-integral and the Aumann integral.

All theorems in this chapter are original.

Chapter V contains our main theorems. In section 1 we show that

the A-integral 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 Yosida-Hewitt 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 set-valued

measure or a set-valued integral, certain natural questions must be

addressed. How do we define additivity of a set-valued function? What

is the proper notion of convergence of sets? Also, can we define

countable additivity for set-valued 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 set-valued 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. 11-12),

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,...,mj-1 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

p-l
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,. . ,Nk-1 have

Nk > Nk- so that if n > Nk
k k-i 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-
m-lb m ) m-1

In this way we obtain a sequence (bm)m=k, where bm Am and
1 m

-m
d(bm,b ) < 2-m

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

n-I
d(b ) < d(bi,bi+)
i=m

n-I
<
i =m

1 1
i m-1
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

p-1
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 set-valued 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 Moore-Smith 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 A-INTEGRAL

The set-valued A-integral will be central in our extension of the

Yosida-Hewitt 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

A-integral of a finitely additive set function.

III.1. Definitions and Basic Properties

In this section we review the definition and basic properties of

the A-integral of a set-valued 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 Yosida-Hewitt 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 A-integrable 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 A-integral 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

SA-integrable over A.

The A-integral, when it exists, shares many of the desirable

properties of the more familiar Bochner integral.

4. Theorem. If i is A-integrable over A, then the integral is unique.

5. Theorem. If 4 and Y are A-integrable

then ai and p. + T are A-integrable over A

fap(do) = a fp(do), Sfi(do) + T(do)
A A A

Although p may not even be additive,

the A-integral 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 A-integrable on both A and B, where A n B = 0,

then i is A-integrable on A u B and

f A(da) = [ftl(d) + fIi(do)]
AUB A B cl

The A-integral 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 A-integral is a countably additive set function in the

sense that if p is A-integrable 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 -k-1
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
-k-1r
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

+ 2-k-1,
+ 2 V,

n Bi) + 2-k-1
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 SA-integrable 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 SA-Integral

The definition of the A-integral 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 SA-integral 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 SA-integrable 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 SA-integrable over E to I. O

2. Theorem. Let i be a real-valued finitely additive set function with

finite total variation defined on E. Then i is SA-integrable 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 SA-integrable over E and thus + +(-1)t_ is SA-

integrable over E. But i = p + (-1)+_; hence i is SA-integrable over

E. O

3. Theorem. Let i be a complex-valued finitely additive set function

with finite total variation defined on E. Then i is SA-integrable over

any E c E, and if t = il + i 2, where ul and '2 are real-valued, 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

Sv-integrable over any E in E by Theorem 2. Thus given any E in Z,

S= + ip2 is SA-integrable 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 SA-integral may be

represented as a double Moore-Smith 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 s-bounded. If

is SA-integrable 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 SA-integrability 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 SA-integrable 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 A-Integral

We start this section wth an existence theorem for the A-integral

of a countably additive set function defined on a a-algebra of sets.

This theorem is the only existence theorem for the A-integral 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 A-integrable

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 interval-valued 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

(2) i is countably additive if and only if a and P are

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

[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 set-valued

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 interval-valued finitely additive set

function defined on E with v(p,S) = M < -. Then i is A-integrable 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 SA-integrable 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 A-integrable 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 set-valued set function defined on E with

values in Rn for some fixed n. We say is rectangle-valued 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 rectangle-valued

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

A-integrable 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 A-integrable 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,
ie-n 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 A-INTEGRAL
AND OTHER INTEGRALS

In the last chapter we defined the A-integral and considered some

of the properties that follow from this definition. In particular, we

noted that the A-integral 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

A-integral differs from the more familiar Bochner and Pettis integrals

in other ways as well; the A-integral 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 X-valued 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 i-integrable function.

2. Lemma. Let g be a simple p-integrable function on S, with i a

complex-valued measure. Define G on I by

G(E) = g(E)(E).

Then G is SA-integrable 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 SA-integrable 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 A-integral, 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 set-valued 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 X-valued t-measurable

simple functions on S converging to the function f i-almost

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 p-almost 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 A-integrability of

the corresponding set function fix.

5. Theorem. Let 4 be a finite positive measure defined on E and let

f:S + X be i-integrable over any A E E. For any A e E define

F(A) = f(A)i(A).

Then F is SA-integrable over any E E E, and

fF(do) = ff dtp.
E E

Proof. Since f is u-integrable on S, there

measurable simple functions (f ) converging

such that

is a sequence of l-

p-almost 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 i-integral of fn is absolutely continuous with respect

to v(i), (2) implies the SA-integral 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
j-Irn 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 SA-integrable. 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 complex-valued measure defined on E, and let

f:S + X be a function which is 1-integrable over any E in E. For any E

in E define

F(E) = f(E)l(E).

Then F is SA-integrable 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 SA-integrable 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 set-valued function with respect to a

single-valued 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 i-integrable

over S (in the sense of Lebesgue) such that f(s) e F(s) u-almost

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 set-valued

function. Then for any E c A define

F(E) = u F(e).
eE

With this definition we can show that when the A-integral

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 A-integrable over E c E, then

fF di fG(dc).
E E

Proof. If the A-integral 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 SA-integrability 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 A-INTEGRAL TO DECOMPOSITION THEOREMS

In this chapter we demonstrate, for a finitely additive set

function L, a relationship between the A-integral of p and the Yosida-

Hewitt decomposition of p. This relationship allows us to extend this

decomposition to the case of finitely additive set-valued set functions.

In 1952, Kosaku Yosida and Edwin Hewitt [24] proved that a bounded

real-valued 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 real-valued set functions defined on an algebra E.

In 1969, James K. Brooks [6] gave a concise, elegant proof which

extended the Yosida-Hewitt 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 Yosida-Hewitt decomposition can be attained

in a constructive manner via the A-integral. In section 1 we treat the

case of scalar-valued set functions, in section 2 vector-valued set

functions, and in section 3 set-valued set functions.

V.1. Scalar-valued 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 real-valued 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

real-valued, we say pi is purely finitely additive if both + and i_ are

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

The next theorem shows that the A-integral is in fact the countably

additive part of the Yosida-Hewitt 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 A-integral 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 SA-integral, 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 real-valued 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

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, real-valued 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 complex-valued 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 complex-valued 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. Vector-valued Set Functions

Building upon the theorems of the previous section, we now extend

the Yosida-Hewitt decomposition to vector-valued 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 X-valued 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 SA-integral of i over

E. Then the countably additive of the SA-integral 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 Hahn-Banach 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 a-algebra 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

The following theorem generalizes the extensions of the Yosida-

Hewitt decomposition given by Brooks [6] and Uhl [23], both of which

provide decompositions for Banach-valued functions. Uhl imposes the

more restrictive condition that the set function be s-bounded (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, X-valued, 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 SA-integrable 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 o-algebra, 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

V.3. Set-valued Set Functions

In this section we consider a direction of generalization of the

Yosida-Hewitt 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 set-valued 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

2. Theorem. Let p be a finitely additive closed interval-valued 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 rectangle-valued

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 A-integral

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 interval-valued set functions.

By Theorem 7.1 in ref. 20, x p is A-integrable 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 < I|x*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

1. Artstein, Zvi, Set Valued Measures, Trans. Am. Math. Soc.
165(1972), 103-125.

2. Aumann, Robert J., Integrals of Set-Valued Functions, J. Math.
Anal. Appl. 12(1965), 1-12.

3. Birkhoff, Garrett, Integration of Functions with Values in a
Banach Space, Trans. Am. Math. Soc. 38(1935), 357-378.

4. Birkhoff, Garrett, Moore-Smith Convergence in General Topology,
Annals of Math. 38(1937), 39-56.

5. Brooks, James K., An Integration Theory for Set-Valued Measures,
I, II, Bull. Soc. Roy. Sci. de Liege (1968), 312-319 and 375-380.

6. Brooks, J.K., Decomposition Theorems for Vector Measures, Proc.
Am. Math. Soc. 21(1969), 27-29.

7. Brooks, J.K., On the Vitali-Hahn-Saks and Nikodfm Theorems, Proc.
Nat. Acad. Sci. 64(1969), 468-471.

8. Brooks, J.K., On the Existence of a Control Measure for Strongly
Bounded Vector Measures, Bull. Am. Math. Soc. 77(1971), 999-1001.

9. Brooks, J.K., and Jewett, R.S., On Finitely Additive Vector
Measures, Proc. Nat. Acad. Sci. 67(1970), 1294-1298.

10. Debreu, Gerard, and Schmeidler, David, The Radon-Nikodfm
Derivative of a Correspondence, Proc. Sixth Berkeley Sympos. Math.
Statist. and Prob. 2(1972), 41-56.

11. Drewnowski, L., Additive and Countable Additive Correspondences,
Roczniki Polskiego Towarzystwa Matematiycznego, Seria I: Prace
Matematycznc 19(1976), 25-54.

12. Dunford, N., and Schwartz, J.T., Linear Operators, Part I:
General Theory, Pure and Applied Math., vol. 7, Wiley-
Interscience, New York, 1958.

13. Hausdorff, Felix, Set Theory, trans. by J.R. Aumann, 2nd ed., New
York, Chelsea Pub. Co. (1962).

14. Hildebrandt, T.H., On Unconditional Convergence in Normed Vector
Spaces, Bull. Am. Math. Soc. 46(1940).

15. Hildebrandt, T.H., Integration in Abstract Spaces, Bull. Am. Math.
Soc. 59(1953), 111-139.

16. Moore, R.E., Interval Analysis, Prentice-Hall, Englewood Cliffs,
N.J., 1966.

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Math. 44(1922), 102-121.

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Soc. 37(1935), 1-20.

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21. Rudin, Walter, Functional Analysis, McGraw-Hill, New York, 1973.

22. Rupp, Werner, Riesz-Presentation of Additive and a-Additive Set-
Valued Measures, Mathematische Annalen 239 (1979), 111-118.

23. Uhl, J.J. Jr., Extensions and Decompositions of Vector Measures,
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Trans. Am. Math. Soc. 72(1952), 46-66.

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 degr-e 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