Convergence theorems for vector integrals

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Convergence theorems for vector integrals
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v, 82 leaves : ; 28 cm.
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Hsieh, Lienzu L ( Lienzu Lin ), 1946-
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Thesis (Ph. D.)--University of Florida, 1981.
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Includes bibliographical references (leaf 81).
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by Lienzu L. Hsieh.
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Typescript.
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Vita.

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CONVERGENCE THEOREMS FOR VECTOR INTEGRALS


BY

LIENZU L. HSIEH














A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1981



















ACKNOWLEDGMENTS


I wish to express my appreciation to each member of

my committee and especially to Professor James K. Brooks

for the continuous guidance and direction of my work toward

the goal of my research. Most of all I wish to thank God in

Lord Jesus Christ for giving me all the grace and strength

to accomplish my research.
















TABLE OF CONTENTS
Page

ACKNOWLEDGMENTS . . ii

ABSTRACT . ..... . iv

CHAPTER
I REAL-VALUED ASYMPTOTIC MARTINGALES INTEGRABLE
WITH RESPECT TO A REAL-VALUED MEASURE 1
1. Introduction ... .. 1
2. Elementary Notations .... 3
3. Conditional Expectation of
a Random Variable .. 6
4. Martingales . .. 7
5. Asymptotic Martingales .. 8
II REAL-VALUED ASYMPTOTIC MARTINGALES INTEGRABLE
WITH RESPECT TO A VECTOR-VALUED MEASURE .. 11
1. Basic Background and Uniform Integrability. 11
2. Decomposition of An Amart .. 27
3. Net Asymptotic Martingales .. 35

III VECTOR-VALUED ASYMPTOTIC MARTINGALES INTEGRABLE
WITH RESPECT TO A VECTOR-VALUED MEASURE 44
1. Basic Concepts and Notations. .. .44
2. Martingales ... . 54
3. Asymptotic Martingales .. 72

BIBLIOGRAPHY . .. . 81

BIOGRAPHICAL SKETCH . ... 82


iii
















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


CONVERGENCE THEOREMS FOR VECTOR INTEGRALS

By

Lienzu L. Hsieh

December 1981


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


In this dissertation we develop several theorems concern-

ing asymptotic martingales, or amarts, in two aspects. First

we treat real-valued asymptotic martingales which are inte-

grable with respect to a vector-valued measure whose values

are in a locally convex topological vector space. Secondly,

we treat Banach-space (or B-space)-valued asymptotic martin-

gales which are integrable with respect to a B-space-valued

measure. In Chapter I, we establish basic concepts concern-

ing real-valued asymptotic martingales integrable with

respect to a real-valued measure. In Chapter II, the uniform

integrability and the decomposition of real-valued amarts

integrable with respect to a vector-valued measure whose

values are in a locally convex topological vector space

are derived. Also convergence theorems of real-valued net

amarts integrable with respect to a vector-valued measure












which takes values in a locally convex topological vector

space are derived. In Chapter III, several theorems con-

cerning B-space-valued martingales integrable with respect

to a B-space-valued measure are developed. We also extend

some of these theorems and an optional sampling theorem to

the amart case.
















CHAPTER I


REAL-VALUED ASYMPTOTIC MARTINGALES



1. Introduction


A sequence of integrable functions (f ) defined on

a probability space (S, Z, p) and adapted to an increasing

sequence of sub-o-fields ( n) is called an asymptotic martin-

gale, abbreviated as amart, if i f ndp < for all nEN and

limit / f dp exists, where N is the set of all natural num-
T CT T
bers, and T is the collection of all bounded stopping times

for (Z ), the limit on T is with respect to the usual order-

ing on T. A martingale is a special case of amart.

The concept of an amart was first given by Meyer [9]

who proved that a continuous parametered scalar-valued amart

converges almost everywhere if it is essentially bounded.

Austin, Edgar, and Tulcea [1] proved that (*) "a real-valued

amart converges almost everywhere if it is L1-bounded."

Chatterji [5] proved that if a Banach space E has the Radon-

Nikodym property, then every E-valued L1-bounded martingale

converges almost everywhere. Chacon and Sucheston [4]

proved that if E is a Banach space with a separable dual and

has the Radon-Nikodym property, then an E-valued amart (fn)

converges almost everywhere in the weak topology of E if the












condition sup f/f |dp < holds. They also showed that
TT

strong convergence need not hold. Smith [111 developed some

convergence properties for weakly measurable Pettis integrable

amarts as well as for strongly measurable Pettis integrable

amarts which take values in a locally convex topological

vector space V. In this dissertation we continue to develop

the convergence properties in a general vector space.

This dissertation is concerned with two topics. The

first topic is the convergence properties for scalar-valued

amarts in which functions are integrable with respect to a

measure which takes values in a locally convex topological

vector space. In this topic amarts are integrable in the

sense of the integral defined by Lewis [8].

In the second topic, we first develop properties which

pertain to martingales whose values are in a Banach space X

and integrable with respect to a measure u which takes values

in a (possibly different) Banach space Y such that there is

a continuous bilinear "multiplication" defined on the product

of these two spaces X, Y and the product lying in a (possibly

different) Banach space Z. After this, we develop these

extended properties to the amart case.

In the first topic, convergence theorems were partially

treated in the dissertation of RobertW. Smith. We will con-

tinue this treatment in three aspects. First we show that

an amart (f ) converges to an integrable function in L1 if

and only if (f ) is uniformly integrable, provided V is












sequentially complete. The proof of the real-valued martin-

agle case can be found in [10, p. 65]. Next we show the

"Riesz decomposition" of an amart into the sum of a martin-

gale and an amart. The proof of the real-valued amart case

can be found in [7, p. 209]. Finally, we show some conver-

gence theory of a net amart which converges in measure under

certain conditions. The proof is developed along the line of

the corresponding theory of scalar-valued amarts developed

in [7, p. 206].

In the second topic, amarts are integrable in the sense

of the integral defined by Bartle [2]. We develop some

properties concerning martingales. Some of the proofs are

developed along the lines of the corresponding theorems of

real-valued amarts developed in [1, p. 18]. After this, we

extend these properties to the amart case. However, we

cannot extend the convergence theorem (*) in this topic,

because it is not true in general that if two P-integrable

functions f and g satisfy A fdy = /A gdp for every set

A in the a-field E, then f = g p-almost everywhere.


2. Elementary Notations


We shall begin by introducing basic concepts and

notations for the real-valued amarts which are integrable

with respect to a non-negative measure X. A field of subsets

of S, or aBoolean algebra of subsets of S, is a non-empty

family of subsets of S which contains the empty set, the











complement (relative to S) of every element, and the union

of any finite collection of its elements. A o-field or

Borel-field of S is a field which contains the union of any

countable collection of its elements. The pair (S, E)

consisting of a set S and a o-field E of subsets of S is

called a measurable space. A function X:EZ- [0, o)is

a countably additive set function on Z if for every disjoint

sequence (E ) is Z with YJ E = E, (E) = E A (E ) where
n=l n=l

the convergence of the infinite series is unconditional.

A function f: S -- R is z-measurable if for every real

value a, the set {seS : f(s) < a} is in E. For convenience,

we will denote this set by (f < a) If f is a E-measurable

function defined on S, then the a-field generated by f is the

smallest o-field of S which contains all sets of the form

(f < a) for any real value a. Similar, the sub-a-field of

Z generated by a family of E-measurable functions (f aA),

denoted by o{(f aEA)}, is the smallest a-field of S which

contains all sets of the form (f B) where B is a Borel

subset of R and a is in A.

A subset E of S is a X-null set if E is contained in

some set F of Z such that X(F) = 0. Any statement concerning

the points of S is said to hold X-almost everywhere, or

simply almost everywhere, if it is true except for those

elements in some X-null set. The phrase "almost everywhere"

is usually abbreviated "a.e." A function f: S -> R is said

to be X-essentially bounded if







5



inf sup f (s) <
N sES-N

where the infimum is taken over all X-null sets N. A func-

tion f: S -> R is a A-null function if (Ifl > a) is a X-nullset,

a > 0. If the function f: S --> R takes only a finite

number of values x,x2,... ,xn and for which the sets

-1
f (x.) = {s: s E S, f(s) = x.} lie in Z, i = ,2,3,...,n,

then any function g from S to R which differs from f by a

A-null function is called a A-simple function. A sequence

(f ) of functions from S to R is said to converge in X-measure,

or converge in measure, to a function f from S to R if

limit {(s: f (s) f(s)I > E) = 0 for every c > 0.
n

A function f on S to R is a A-measurable function on S if

there is a sequence of X-simple functions converging to f

in measure.

Two other convergence concepts for sequences which are

often used in real-valued measurable functions are almost

everywhere convergence and convergence in mean. We say that

a sequence (f ) converges to f almost everywhere, fn f a.e.,

if there is a A-null set N such that limit f (s) = f(s) for
n~oo
n-*co

every s in S N. Let (f ) be a sequence of L1-integrable

functions, we say that (f ) converges to f in mean if the

sequence (f ) converges to f in L1-norm. This means

limit /If -f d\ = 0. If A(S) = 1, then (S, E, A) is called
n

a probability space. On a probability space, a E-measurable












function will be called a random variable, abbreviated as r.v.

A random variable is also a measurable function on that

probability space.


3. Conditional Expectation of a Random Variable


From now on we will use (S, Z, p) to denote a probabil-

ity space. For any sub-a-field J of Z, the countably additive

set function v: J -> R is said to be absolutely continuous

with respect to P, denoted by v << P if v(E) -- 0 whenever

P(E) -> 0. Now we state the well-known Radon-Nikodym theorem:

Let (S, Z, X) be a finite positive measure space, and

v a finite positive measure on Z, which is absolutely con-

tinuous with respect to X. Then there exists a unique func-

tion f in L1(S, Z, X) such that v(E) = IEfdX for each E in Z.

Let f be a random variable defined on S such that

/f fdp < -, and J be a sub-a-field of E. Then the set func-

tion v : J -- R defined by v(E) = /Efdp for E E J is a

countably additive set function and is absolutely continuous

with respect to p. By the Radon-Nikodym theorem, there is

a J-measurable, integrable function g such that E fdp = fEgdp

for every E in J. A J-measurable function which differs

from g by a null function is called the conditional expectation

of f given J, and is denoted by E(flJ). The conditional

expectation of f given by (f : a E A) means the conditional

expectation of f given the sub-a-field of Z generated by

(f : a E A), denoted by E(f/f : a E A). Intuitively











E(f/f : a c A) is the "best estimate" of the random variable

f given information from the random variables (f : a A}.


4. Martingales


Let (f ) be a sequence of random variables and (Zn)

an increasing sequence of sub-a-fields of Z. We say that

(f ) is adapted to (Z ) if f is a Z -measurable function
n n n n
for each n in N. A sequence of random variables (f ) which

is adapted to (Z ) is called a martingale if f|fn dp < m

for each n E N and E(fn /Zm = f for each m,n e N, n 2 m

or equivalently, /Efndp = fEfmdp for each E in Zm, n,m E N

and n a m. A martingale is said to be L -bounded if

sup /Ifn dp < c.
n

For any martingale (fn, Zn, n E N) the following

conditions are equivalent:

(a) The sequence (f ) converges in L .

(b) There exists an integrable random variable f

such that f = E(f/Z ) for all n in N.

(c) The sequence (f ,n n N) satisfies the uniform

integrability condition, that is,

limit / ( f >a) fn dp = 0 uniformly in n N.


The proof of the preceding property can be found in [10,

p. 65]. A martingale is called a regular martingale if it

satisfies one of these equivalent conditions.












Let (f En, n E N) be a sequence of random variables

adapted to an increasing sequence (Z ) of sub-a-fields of Z.

We call (f ) a submartingale if I fn dp < m for each n and

E(f /Z ) > f or ./f dp /If dp, for n,m N, n > m, EE Z
n rn m E n Em m

(fn) is a supermartingale if f/f ndp < m for each n N

and E(f /Zm) < fm, or IEfndp IEfmdp, for n,m E N, n m, E Zm

Note that (f En, n E N) is a supermartingale if and only

if (-f Zn, n E N) is a submartingale. We say that a sequence

of random variables (Zn, n e N) is an increasing process if

it satisfies the conditions

(1) Z = 0; Zn < Zn+ for n 2 1;

(2) E(Z ) < for each n, where E(Z ) = Zn dp.

We state the well-known Doob's decomposition of positive

supermartingales:

Every finite positive supermartingale (f n E N) can be

written in one and only one way as the difference between

a finite positive martingale (gn, n E N) and an increasing

process (h n E N), and f = g h for each n c N.

The proof of this property can be found in [10, p. 171].


5. Asymptotic Martingales


Throughout this section we will continue to assume that

(f ) is a sequence of random variables adapted to an increas-

ing sequence (n ) of sub-o-fields. A random variable T: S -> N

is a stopping time for (Z ) if (T=n) E Zn for each n in N.












Let T be the collection of all bounded stopping times for

(Z ). For each T E T, we define a random variable f by

f (s) = f (S)(s), that is, f (s) = fn(s) for each s in the

set (T=n), n E N. Let TIT2 be two bounded stopping times.

We say that T1 is not less than T2, denoted by T1 ; T2'

if T (s) 2 T2(s) almost everywhere. A partially ordered set

(A, f) is said to be directed if every finite subset of A has

an upper bound.

(fn, n, n E N) is an asymptotic martingale, or simply

amart, if /fn dp < m for all n E N and limit f Tdp exists.
TET

A martingale (f Zn, n E N) is also an amart. This fact
n n
is proved in the following manner:

Let T be any bounded stopping time for (Z ). Then there

is an integer r such that T < r. Therefore

r r
if dp = Z f dp = f dp = If dp
n=l (T=n) n= (T=n)

= Ifldp.

Hence If dp does not depend on the choice of T e T. Thus

f(n) is an amart. Similarly submartingales and supermartin-

gales are amarts if (If dp, n E N) is bounded.

As in Doob's decomposition of positive supermartingales,

there is a decomposition for amarts given as follows:

Let (f n E N) be an amart. Then fn can be
n n n
uniquely written as f g + h where (gn, ,n e N), or

simply denoted by (gn) if (Zn, n E N) is fixed, is a marting.le.







10



and (h ) is an amart with h -- 0 in L In addition,

(h T c T) is uniformly integrable and h -- 0 a e. For

the proof of this decomposition theorem, see [7, p. 209].
















CHAPTER II


REAL-VALUED ASYMPTOTIC MARTINGALES INTEGRABLE
WITH RESPECT TO A VECTOR-VALUED MEASURE



1. Basic Background and Uniform Integrability


In this chapter we shall continue to develop the theory

of real-valued amarts which was treated partially in [11].

These amarts are integrable in the sense of the integral

defined by Lewis [8]. Lewis developed the integration theory

chiefly through the study of the P-semi-variation of the

vector measure whose values are in a locally convex topo-

logical vector space V, where P is a semi-norm on V.

We shall start by introducing the theory of integration

developed by Lewis, and the properties of real-valued amarts

which are integrable with respect to a V-valued measure p,

investigated by Smith in [11].

Throughout this chapter (S, Z) will denote a measure

space, V a locally convex topological vector space, and j

a V-valued countably additive set function on Z. A contin-

uous function P from V into R satisfying

(1) 0 < P(x) < + c,

(2) P(ax) = laP(x),

and (3) P(x+y) P P(x) + P(y)

for any x, y V and any complex number a, is called

11












a continuous semi-norm P defined on V. If x* e V* and P

is a continuous semi-norm defined on V we will write x* < P

whenever Ix*(x) < P(x) for all x E X. If P is a semi-norm

on V, then the P-semivariation of p is the function from Z

into the extended reals defined by

l ID(E) = sup {v(x'*1,E): x* P}

where v(x*u,-) is the scalar variation of x*p, that is,

n
v(x*p,E) = sup Z Ix*p (Ei)
i=l

where the supremum is taken over all finite sequence {Ei}

of disjoint subsets of E in E.

For proofs of the next three results, see Lewis [8].

Proposition 2.1-1 [8, p. 158]. If P is a countably

additive measure and P is a continuous semi-norm on V, then

|lp i (*) is monotone, countably subadditive, real valued, and

P[p (E)] i | fl p (E) 5 4 sup {P[i(F)]: F
Theorem 2.1-2 [8, p. 158]. If u is a measure, P a con-

tinuous semi-norm and (E n N) a convergent sequence

of sets in Z, then

||lp (limit E ) = limit Hl l(En)
n n
Corollary 2.1-3 [8, p. 158]. If p is a measure and

(En, n E N) is a convergent sequence in Z, then


i(limit En) = limit i (En) .
n n












A subset of S is called a p-null set if it is a subset

of a set E E Z such that l|p| (E) = 0 for every continuous

semi-norm P. Any statement concerning the elements of S

is said to hold p-almost everywhere, or simply almost every-

where if it is true except for those elements in a u-null

set.

A p-simple function f: S ->- R is a function which is

u-almost everywhere equal to a function

n
g: S -> R, g(x) = Z ciIE (x)
i=l i
n
where c. R and E. E Z for each i = i,2,...,n, E. = S
1 1 i=l 1

and I (*) is the characteristic function of E. A function
1
f : S -> R is p-measurable if there exists a sequence of

W-simple functions which converge to f p-almost everywhere.

A function f : S -> R is p-integrable if

(1) f is x*p-integrable for each x* E X* and

(2) for each E E there is an element of X,

denoted by f fdv, such that x*( EfdO) = Efd(x*i) for each

x* in V*. Since the topology on V is Hausdorff the integral

is well-defined. It is obvious that the integral is linear
n
and for every simple function f(x) = c IE. f is p-inte-
i=l i
n n
grable and E( Z cIIE ) dp- Z c.i (E n E.) for E E E. If f is
i=l 1i i=l

bounded and u-integrable, then

P[ Efdu] < l| Ip (E). sup {f(s): s S}


for each E E Z and continuous semi-norm P.











Theorem 2.1-4 [8, p. 160]. (1) If f is p-integrable,

then the set function on Z defined by 4 (E) = fEfd is a

measure,

II || (E) = sup { IE f(t) v(x*p,dt) : x* < P}

and limit |(| H (E) = 0 for each continuous semi-
1 llp(E )0

norm P.

(2) Let (f ) be a sequence of p-integrable functions

which converge pointwise to f on S and g be a p-integrable

function such that Ifnl g for each n. Then f is u-integrable

if V is sequentially complete. If f is p-integrable, then

IEfdp = limit E f nd uniformly with respect to E E Z.
n
In Theorem 2.1-4 we can replace the hypothesis (fn)

converges pointwise to f on S" with "(f ) converges to f

i-a.e. on S." The result is still true, because for each

x* E V* there is a continuous semi-norm P on V such that

x* < P. This P can be chosen as

P: V -> R+, P (x) = x* (x) for each x e V. Therefore f is

x*p-integrable for each x* e V*. The rest of the proof is

the same as the proof in [8, p. 160].

Definition 2.1-5 [11, p. 60]. If (f n n e N) is

a sequence of p-measurable functions adapted to an increasing

sequence of sub-a-fields (En) of Z, then (fn' En, n N) is

defined to be an amart if

sup {/ fn dv(x*) : x* < P} <

for each n E N and each continuous semi-norm P, and

limit fdi exists in V. The existence of this limit means
TET











that there is a vector x V such that for each e > 0 and

each continuous semi-norm P there is a o E T such that if

T > u then sup { x*(ff dx x)I : x* < P} < e.

Lemma 2.1-6 [11, p. 61]. Let (f ) be a sequence of

P-integrable functions adapted to an increasing sequence of

sub-o-fields (Z ) of Z, and T (E) = IEf Td for each T E T.

If sup {|11 (S)II : T E T} < for every continuous semi-norm

P, then for X > 0

lU p ({s S: sup If (s)I > XI) s (sup {fIfT dv(x*p):
n
TE T, x* P ) for every continuous semi-norm P.

Proposition 2.1-7 [11, p. 62]. If (f, n Z, n E N) is

an amart then sup { /f d(x*)j : x* < P, T E T} < m

for every continuous semi-norm P.

Proposition 2.1-8 [11, p. 65]. For each E EZ, the

restriction of the sequence (f ) to E is an amart for

(Z n E : n E N), where Z n E denotes the o-field consist-

ing of all intersections of elements of En with E.

Definition 2.1-9 [11, p. 69]. A sequence (f ) of

p-integrable functions is said to be Ll-bounded if

sup {/ fn dv(x*Y) : x* < P; n E N} <

for every continuous semi-norm P.

Theorem 2.1-10 [11, p. 64]. If V is a sequentially

complete space and if (f n N), (g E n N) are
n n n n
amarts, then (fn V gn, En, n N) and (fn A gn, ,En n N)

are also amarts, where (fn V gn) (s) =max {fn(s), g (s)} and

(fn gn (s) =min {f (s) g (s)} for each s in S.











Proposition 2.1-11 [11, p. 70]. If V is a sequentially

complete space and if (f Zn, n E N) is an L -bounded amart,

then:

(a) ( f Z n N), (f Z n E N), (f, Z n E N)
n n n n n n
and (-a V f A a, Z n E N) are also L -bounded

amarts, where a 2 0.

(b) Sup {[ If I d(x*p) I: T E T, x* < P} < c.

(c) Sup { f I: n e N} < u -almost everywhere.


Proposition 2.1-12 [11, p. 72]. Let (f ) be a sequence

of i-measurable functions adapted to (Zn, n e N) and suppose

there is an integrable u-measurable function g such that

If g for all n E N. Then if V is sequentially complete,

the following are equivalent:

(a) (f Zn, n E N) is an amart;

(b) the sequence (fn) converges almost everywhere to

a u-integrable function f.


Lemma 2.1-13 [11, p.73]. Let (fn' Zn, n e N) be an

amart and assume that V is sequentially complete. Then the

sequence (fn) converges to a u-integrable function f u-a.e. if

sup { fn dv(x*u): n E N, x* P} < c

for every continuous semi-norm P.

The next theorem constitutes an "optional sampling

theorem" for amarts.











Theorem 2.1-14 [11, p. 74]. Let (f En, n E N) be

an amart and (Tn, n E N) be a non-decreasing sequence of

bounded stopping times for (Z n e N). Define g = f and
n
S= Z = {B E Z: B('{T = n} CE n e N}.
k T k n

Then (g ', n E N) is an amart if V is sequentially complete.
n n

Lemma 2.1-15 [11, p. 76]. If (f ,Z n E N) is an

amart and a is a stopping time, then (fn n E N) is an

amart if V is sequentially complete.


Proposition 2.1-16 [11, p. 77]. Let V be a sequentially

complete space. If (f n n E N) is an amart and (Tk, k E N)

is an increasing sequence of bounded stopping times with

Tk > k, and if for each continuous semi-norm P

sup {/(sup If f k ) dv(x*p): x* < P} < c,
kEN k

then (f ) converges a.e. on the set B = {suplfnI < c}.
n
Now we continue our development of the theory of amarts

which has been illustrated in preceding paragraphs.


Definition 2.1-17. Let (f ) be a sequence of p-integrable

functions and D (E) = Ef ndp for each n E N, E E Then
n
(f ) is defined to be uniformly absolutely continuous with

respect to 1 if

limit lIn Ip (E) = 0
1 1| (E) O n
p
uniformly in n, for every continuous semi-norm P, where

Inl1 (E) = sup {/ f dv(x*p): x* P}.
np E n











Definition 2.1-18. Let (f ) be a sequence of p-integrable

functions. Then (f ) is defined to be uniformly integrable

if limit II ll (I fn > N) = 0 uniformly in n, for every con-
N np n
tinuous semi-norm P.

The next theorem shows that uniform integrability is

equivalent to L1-boundedness and uniform absolute continuity

for a sequence of U-integrable function.


Theorem 2.1-19. Let (f ) be a sequence of p-integrable

functions. Then the following are equivalent:

(a) (f ) is L -bounded and uniformly absolutely

continuous with respect to p;

(b) (f ) is uniformly integrable.


Proof. First we shall show that (a) implies (b).

Let c be an arbitrary positive number and P an arbitrary but

fixed continuous semi-norm on V. Since (f ) is uniformly

absolutely continuous with respect to i, there is a number

6 > 0 such that ||nl~ (E) < whenever E E Z and IIwjI (E)< 6.

Since (fn) is L1-bounded, define

M = sup {/ fn dv(x*O): x* P, n E N},

then M is finite.

Choose an integer N such that N > M/6. We claim that

llp (If n>N )<5 holds for each n N. If this is not true,
then there is some integer no such that liVllp(Ifn > N0) 2 6.
0










Since

Iull (I f > N ) = sup {v(x*T ( fn > N )): x* < P},
S 0 0

there is some x* e V*, x* < P such that v(xv*p,(Ifn > No))26.
0 n 0 0
0
Then
> f Idv(x*U) 2 No v(x* ,(If >N )) > M.
(Ifn >N ) o o

This is impossible because
M = sup {/Ifn dv(x*p): x* < P, n E N}.
Therefore IIlp (fn > N ) < 6 is true for each n e N.

This implies iinl |(Ifn > No) < e for each n E N.
Thus we have proved that (a) implies (b). Next we want to
show that (b) implies (a). We still let P be an arbitrary

but fixed continuous semi-norm.
Since (f ) is uniformly integrable,

limit I| II (If l > N) = 0 uniformly in n.
N-+ p n

Given c = 1 there is an integer N1 such that

[I n p (( Ifn > N1)) < 1 for each n E N where

I Ip ( (fn N > N ) = sup { if nl fn dv(x*u): x* < P}.

Then

flfn dv(x*p) = (If >N) I dv(x*i)+ (I f N 1 )fn dv(x*p)

< 1 + N *v(x*p, (fn
< 1 + N1I [lp(S), for each n e N and each x* < P.

It follows from Proposition 2.1-1 that ~ Ilp (S) is finite.











Therefore

sup {/fn dv(x*p) : n E N, x* < P} < 1 + N p [ (S) < m.

This means that (f ) is L -bounded. Let E > 0 be given.

Since (f ) is uniformly integrable, there exists an integer

N such that

||( f I> N ) < for each n e N.


Choose 6 = 2- and let E be in E such that IIjl(E) < 6.

Then

EIfn dv(x*p) = fEn( f N f dv(x*Y)
En En(if >N) n ^
+ {E (jf n N)Ifn Idv(x*p)


( I I fN N fn dv(x*p) + N .v(x*p,E)< + N IHL p(E)< E

for each n N and for each x* V*, x* P.

Therefore

[ln pl(E) = sup {E fn dv(x*p): x* < P} < E

for each n N. This means that given any E > 0, there is

some 6 > 0 such that in Ipl(E) < for each n N, provided

that E and 1lI (E) < 6. Therefore (fn) is uniformly

absolutely continuous with respect to p.

Definition 2.1-20. Let (f ) be a sequence of p-integrable
functions. Then (fn) is said to converge to a y-integrable

function f in L1 if for every continuous semi-norm P the

following is true:
limit fn -fdv(x*p) = 0 uniformly in x* V*, x* < P.
n f










Note that from Proposition 2.1-1 and 2.1-4 it follows that

if f is p-integrable and (E) = }Efdp, then II | (E) < m

for each E in E and each continuous semi-norm P.


Lemma 2.1-21. Let (f ) be a sequence of p-integrable

functions. If (f ) converges to a p-integrable function f

in L then (f ) is uniformly integrable.

Proof. Let P be an arbitrary but fixed continuous

semi-norm. Since (fn) converges to f in L ,

limit f -f[dv(x*0) = 0 uniformly in x* E V*, x* < P.
n J

Take o = 1, then there is an integer n1 such that

f -fnfdv(x*p) < 1 for each n 2 nI, n E N and x* P.

Hence

Iffn dv(x* ) I fn -fldv(x*p) + If dv(x*w)

< 1 + ( llp(S) for each n E N, n > nl, where

S(S) = sup { |f dv(x*p): x* < P}.
Therefore

sup { fn dv(x*y): x* P, n N}
= max {1 + ||| (S) 1 + | l (S) ,

1 + I p(S) ,... ,l + (nl1 n(S)< ,
where

|l|ki p(S) = sup {ifk dv(x*) : x* P}

for each k = 1,2,...,nl-1. This shows that (fn) is L1-bounded.







22



Let E > 0 be given. Since (f ) converges to f in LI,

we have,

limit If -fldv(x*u) = 0 uniformly in x* < P.
n '

There is an integer no such that

If -fdv(x*p) < for each n e N, n > n and each
x* P. Since f is w-integrable, there is a number do > 0

such that | ||p(E) < whenever E E Z and Ivlj (E) < 6S

Now,

IE fnldv(x*u) < Elfn-f dv(x*Y) + jE fdv(x*P)

E f n-f dv(x*i) + | p(E) for each x* < P,

therefore for every integer n, n 2 no and every E in Z with

i ijL (E) < 60, we have

fE fndv(x*p) < for each x* P.

Also because fl"',,fn _- are p-integrable, there exist
0
numbers 6',6 ,...,6 -i such that for each m {1,2,... ,no-l,
0
IIm p(E) < E whenever E E Z and Iil < 5m Choose

6 = min {6o ,1,...,6n _-}. Therefore we have
o

f fndv(x*i) < E for each x* < P and each n E N, whenever

E c 7 and |lllp (E) < 6. This means (f ) is uniformly abso-

lutely continuous with respect to i. It follows from the

preceding theorem that (fn) is uniformly integrable.

Note that if (f ) is an amart, then for each n E N,

f is M-integrable. This fact follows from the definition
n
of amart and Proposition 2.1-8.












Definition 2.1-22. Let (f ) be a sequence of p-measurable

functions. Then we say (f ) converges to a p-measurable func-

tion f in p-measure, or simply in measure, if for every E > 0,

limit i|P l1 ({s E S: If (s)-f(s) >C}) = 0
n

for every continuous semi-norm P.


Definition 2.1-23. Let (f ) be a sequence of p-measurable

functions. Then we say (f ) converges to a p-measurable func-

tion f U-uniformly if for every E > 0 and every continuous

semi-norm P there exists a set A in E such that

Ifl (S-A ) < and (f ) converges to f uniformly on A

The next theorem extends the Egoroff theorem to the case

of a measure space (S, Z, p) where p takes values in a locally

convex topological vector space V.


Theorem 2.1-24. Let (f ) be a sequence of p-measurable

functions from S to R. If (f ) converges to a u-measurable

function f p-a.e., then (f ) converges p-uniformly to f.


Proof. Let > 0 be given and let P be a continuous

semi-norm. Suppose that E is the p-null set such that

(f (s)) converges to f(s) for s V E. Let

E = {s S: f (s)-f(s) < 1 for r 2 k},
k,m r m

then Ek+l,m E Ek,m. Since (f (s)) converges to f(s) for
00
s V E, we have U E = S E for all m. Therefore
k=l












E= ( (S-E ) = limit (S-Ek ).
k=l k

It follows fromTheorem 2.1-2 that

|uIp (E) = limit I111 (S-Ek,m)
k
Since ll|p(E) = 0, there is an integer km such that

II Ip(S-Ek < -for each m.
k ,m m

Let A = E then S A = l (S-E ) .
m=l m=l m

It follows from Proposition 2.1-1 that


l (S-A ) l(S-Ek ,m) Z = :
P m=l m m=l 2

Given any 6 > 0, there is an integer mo such that

1
S< 6. If s A then s lies in E also.
m E k ,m
Io

Therefore If f(s)-f(s) < m < 6 for each k 2 k
k m m
o o

Thus (f (s)) converges to f(s) uniformly on A This

proves that (f ) converges to f p-uniformly.


Lemma 2.1-25. Let (f ) be a sequence of u-measurable

functions from S to R. If (f ) converges to a p-measurable

function f n-uniformly, then (f ) converges to f in p-measure.


Proof. Let E > 0 be given and let P be a continuous
1
semi-norm. Then there is an integer m such that E > -
m

Since (f ) converges to f p-uniformly, given any number

6 > 0, there is a set A in Z such that Ip||,(S-A0) < 6











and (f ) converges to f uniformly on A Consequently, there

is an integer k such that fk (s) f(s)1 < for each k 2 k
m k m m
and each s A Since
1
{sES : Ifk(s)-f(s) I>E}C {sS: Ifk(s)-f(s) 2 m =


S {sES: fk (s)-f(s) < -1}S-A for each k 2 k
k m 6 m

Therefore HpI {s: Ifk(s)-f(s) I > E} l p| (S-A6)< 6

for each k 2 k

Thus limit l l is: fk(s)-f(s) > } = 0.
k

This proves that (f ) converges to f in p-measure.


Theorem 2.1-26. Let (f ) be a sequence of n-measurable

functions. If (f ) converges p-a.e. to a p-measurable func-

tion f, then it converges to f in p-measure.


Proof. The result obviously follows from Theorem 2.1-24

and Lemma 2.1-25.


Theorem 2.1-27. Let (fn, n, n E N) be an amart and

let V be a sequentially complete space. Then the following

are equivalent:

(a) (f ) converges to a p-integrable function f in L ;

(b) (fn) is uniformly integrable.


Proof. Since (f n n e N) is an amart, we have that

(f ) is a sequence of p-integrable functions. It follows

from Lemma 2.1-21 that (a) implies (b). We need to show

that (b) implies (a). Since (f ) is uniformly integrable,










by Theorem 2.1-19, we have that (f ) is L1-bounded. Also

(f n, n e N) is an amart. Therefore (f ) converges to
a w-integrable function f y-almost everywhere from
Theorem 2.1-13. Therefore (f ) converges to f in i-measure.
Thus for any E > 0,

limit |1 IL({SES: If (s)-f(s) > ) = 0
n pn 3 H iI (S)
for each continuous semi-norm P.

(fn) is uniformly integrable; hence (fn) is uniformly
absolutely continuous with respect to u; that is,
limit 1n1 (E) = 0 uniformly in n N, where
I Ij p (E) O
P
n (E) = Ef du for each n e N and E E Z. Therefore there

is a positive number n1 such that for each n c N,

|lOnl (E) < whenever E E Z and |lpp (E) < n1. Since f is

i-integrable, we have limit i I11 (E) = 0 where
II ll' p (E)-0
(E) = IEfd for each E E E. Therefore there is a positive
number n2 such that II p(E) < whenever E Z and
2 p 3

il p(E) < n2. Define q = min { ,n 2}. Then there is an
integer no such that

liu p({sES: fn(s)-f(s) > 3 l (S)

for each n N, n > no. Define

A = {scS: f n(s)-f(s) > 3[|.IIp(S)
P











Therefore

I fn-f dv(x*p) = _S-A nfn-f dv(x*p) + fA nfn-fdv(x*0)
n fn d

3||p (3) I' (S-A ) + IAIfn dv(x*Y) + IAnfdv(x*p)
31pIj p (S) p n A n A


< + I (A ) + IIl p(A ) < for each n c N,
3 np n p n

n > n and each x* < P, x* E V*.

This proves that limit I f-f dv(x*p) = 0 uniformly in
n-oo
x* < P, x* E V*. Thus (f ) converges to f in L .



2. Decomposition of an Amart


In [7, p. 209] Edgar and Sucheston showed that a real-

valued amart (Xn) for (Z ) integrable with respect to a

probability measure P can be expressed as X = Y + Z ,

where Y is a martingale and Z is an amart with Z -> 0
n n n
in L In this section we develop this property for the

real-valued amart (f ) for (Z ) integrable with respect to

a V-valued measure p, where V is a locally convex topological

vector space, provided V has the Radon-Nikodym property and

some additional properties.

Let (S,Z) be a measure space p: S -> V be a countably

additive set function. We need to use the next theorem in

the proof of the decomposition theorem for an amart.


Theorem 2.2-1 [6, p. 165]. Let (S,Z ,) beameasure space

on Z. Let Z1 be a sub-o-field of Z, and p, be the restriction











of w from E to 21. Then

(a) convergence in pw-measure implies convergence in

p-measure;

(b) a p -null function is a p-null function;

(c) a p -null set is a p-null set;

(d) a p -measurable function is i-measurable;

(e) a pl-integrable function f is p-integrable and


E f d = IEf di for each E E 1.

Since a function is p-integrable if and only if it is

v(p)-integrable, the result (e) of Theorem 2.2-1 still holds

if A is a real-valued measure.


Definition 2.2-2. A sequence of functions (f ),

f : S -> R for each n e N, adapted to an increasing sequence
n
of sub-o-fields (Z ) of Z is defined to be a martingale if

the following hold:

(1) (fn) is a sequence of p-integrable functions.

(2) Sup { If dv(x*l): x* i P} < for each n E N and

for every continuous semi-norm P.

(3) Given any n E N and any E E n


Ef nd = Ef md holds for each m E N, m > n.

Definition 2.2-3. Let V be a locally convex topological

vector space. Then V is said to have the Radon-Nikodym

property if for every V-valued measure space (S, Z, p) and

every V-valued measure v such that











limit IvjI (E)=0 and IIvI (S) < for every continuous
l I I (E)*0 P
semi-norm P, there is a p-integrable function f from Z to R

such that v(E) = EfdU for every E in E. Note that throughout

this section we will assume that V is sequentially complete

and has the Radon-Nikodym property.

Let (f Z n e N) be a real-valued amart in the measure

space (S, Z, p), where p is a V-valued measure. Since (fn)

is an amart, (f ) is a sequence of v-integrable functions.

Define a function vn: Z -- V by un(E) = Ef nd for each E E

and each n e N then limit v" (E) = 0 and |lvli (S) < from
I-lp (E)-O P p
Theorems 2.1-1 and 2.1-4. Define a function v : Z V by
m m

k fk k
v (E) = fkdp for k, m E N and each E E Z then V' is
m E k m m

is a countably additive set function on Z Since

limit Ik 1 (E) = 0 for each continuous semi-norm P,
I| p (E) 0 p
k
we have limit mI pl (E) = 0. Now
| (E) -0
p

n n
kn k n f
v(x*k ,S) = sup Z x*v (E) = sup f dx*
m E. E i=l m i E.Z i=l k
i m i m

t t
Ssup E fkddx*p = sup Z |x*vk (F)
F.Z j=l ij F.cZ j=l

k k k
= v(x*k S), hence k lv (S) Ilvkl p(S)< oo
mp p

Since V has the Radon-Mikodym property, there is a
k
,/Zm-integrable, Z m-measurable function gm such that
m ~mm











k k
(E) = g di for each E E .
m Em m
r d k
But E kd = m (E) for each E E Z therefore
JE k m m


E m = E kdp for each E m-

It follows that for m, k E N we have


SE d = E kdp, n N, n > m, E m

Lemma 2.2-4. Assume that V is sequentially complete and

has the Radon-Nikodym property. Let (fn, n, n E N) be
k
an -bounded amart and gm = E(fk/Zm) be given as before,

then (g Zk = k E N) is an L -bounded amart and it
m m m 1
converges to a p/ m-integrable function gm y/Z-a.e.


Proof. Since (fn, n, n N) is an amart it follows

that limit f Tdi exists in the sense that there is an element
TET
v in V such that
o
limit sup { x*(ff dpi-v ) : x* L P} = 0
TE T

for every continuous semi-norm P. Let T be the collection of
k
all bounded stopping times for (Em, k E N) for each m, that


t
is, (T = t) Z for each t E N. If T is a bounded stopping

time in T with T m, then (T = t) E Z C Z for each t i m;
m m t
therefore T is also a bounded stopping time in T. Let the

integer r be an upper bound for T, then

r r
g tdy = E g mdp = E ftdt = f dp.
f m t=m (T=t) t=m (T=t) T












Therefore,

limit sup { x*( IgTd-vo) : x* P}
TET ) m o
m

= limit sup { x*(Ifd d-v ) : x* P} = 0
TET

for every continuous semi-norm P. Also

sup {/lgmk dv(x*) : x* < P} = mki p(S)

E 1v klb(S) < is true, therefore
k k
(g = k N) is an amart. Since (f Z n N)
m m m n n
is L1-bounded,

sup {/fg" dv(x*u): x* 5 P, n e N}

= sup {/ fn dv(x*) : x* < P, n E N} < .

k k
Consequently, (g Zkm k E N) isan L -bounded amart.
m m m 1
Therefore (g k c N) converges to a /E m-integrable,

Z -measurable function g I/Z -a.e. This proves the lemma.
m m m
Now let g be the p/Z -integrable, Z -measurable func-
m m m
tion given in the preceding lemma. Then gm is x*/p-integrable
k
from Theorem 2.2-1. Since (g k E N) converges to g
mm
p/Em-a.e. it follows, that

i| Zml p({sES : gm(s) -o g (s)} ) = 0

for every continuous semi-norm P. Define

A = {sS : g (s) -* gm(s)} then v(x*p/Zm,A) = 0 for each
mm
x* E V* and A E m; hence 9(x*U,A) = v(x*/Zm ,A) and

IIp b(A) = 0. This proves that (g k E N) converges to gm

p-a.e.











Theorem 2.2-5. Let (f Zn, n N) be an L -bounded

amart and gm be a /Z m-integrable function such that

(E(fk/Zm) : k e N) converges p-a.e. to gm. If for each

m N, gm is i-integrable and if there is a p-integrable,

E1-measurable function g such that |E(fk/Zm) < g for

m, k E N the the following are true:

(1) (gm, Zm, m E N) is a martingale.

(2) (fn, n n N) can be uniquely expressed

as the sum of a martingale and an amart.

The uniqueness is in the sense that if fn = g + h =

gn + h' where (g ),(g') are martingales and (h ),(h') are
n n n n n n
amarts, then g = g' p-a.e. and h = h' p-a.e. for each

integer n.

Proof. (1) Since gm is p-integrable it follows that

sup {/ gm dv(x*0) : x* < P} < -

for every continuous semi-norm P. Let n be an arbitrary

integer and E Then E c Z for each m 2 n, and
n m

fEgmdV = limit IE E(fk/Zm)dv =

limit fkdu = limit E E(fk/ n)dv = E gndp
k k
from Theorem 2.1-4. This shows that (g Em, m E N) is a

martingale.

(2) Let h = f g for each n N. Now
n n n
Shn dv(x*u) = If n-g dv(x*p)

(Ifnl + Ign ) dv(x*p)

= fndv(x*i) + fIgn dv(x*0).











Therefore,

sup { fh dv(x*) : x* < P
r *
s sup { fn dv(x*:) : x < P}

+ sup { Ign dv(x*u) : x* < P} < c.

Let T be a bounded stopping time in T, and T is bounded

by an integer r, then

f r r
Tdli = (T=t) gtd t = (T=t)grd


= grd1 = g1di.

Since limit f di exists, there is an element v of V such that
T ET
limit sup { Ix* ( f d-v) : x* P =0
TcT f
TET

for each continuous semi-norm P. Let vo = v gld then

v E V. Therefore for each x* V*, x* < P we have
o

x*( h _d-vo) = x*(f(fT-gT)di-vo)


= (f,-g,)dx* -x*(vo) ={fdx*i gTdx*p x*(v0)

= x*( ffdu) x*([g dp) x*(v0) = x*(jfTdp)


x*( gldp) x*(vo) = x*( fdu- jgldp-vo)


= x*( f dp v).

Consequently,

limit sup {|x*( h d-vo) : x* P}
TT

= limit sup {I x*( f dl-v) : x* < P} = 0.
TCT T











Therefore, (h E n E N) is an amart. Assume that f = h'+g

where (h', E n E N) is a martingale and (g' E n N) is

an amart. Let n be an arbitrary integer and E n, and x* be

an arbitrary element in V*, then

IE gmd x*i = IE gnd x*i

for each m n, m E N, and

limit fE f d x*p = limit E(f /Z )dx*u
m fE m m E m n
m m


= IE limit E(fm /Zn)dx* = IE gn dx*V

from the Lebesque Dominated Convergence Theorem. Consequently,

limit h d x*F = limit f d x*w limit E gd x*p
i E m m fE m m fE m
m m m

= E gd x* IE gnd x* = 0

for each E EZ and for each n N and each x* E V*. Similarly,
n

limitE h'd x* = 0 for each x* E V* and E E n N.
E m n
m
Since h + g = h' + g' for each m E N,
m m m m

limit h d x*u + limit g dx*,
E m jE m
m m

= limit E h'd x*p + limit E g'd x*d for E E n c N.
m m
Therefore

limit g d x*p = limit ( g'd x*p
mm m mE

for EE E, n N and x* V*. But limit Egmd x*p = E gnd x*y
n' em

and limit gd x*p = g'd x*p for E E n N and
m E E En
each x V*; therefore
each x* V*; therefore












IE gndx* = JE gndx*

for each E c Z and each x* E V*. It follows that
n
g = g' x*/Z-a.e. because g ,g' are Z -measurable.

From Theorem 2.2-1 we have g = g' x*i-a.e. for every

x* C V*. Consequently, g = g' p-a.e. for each n e N.

This implies that h = h' i-a.e. for each n EN. This

completes the proof of the theorem.



3. Net Asymptotic Martingales


Definition 2.3-1. Let A be a directed set and

(f a a A) be a net of u-measurable functions from S to R

adapted to an increasing net (Z a E A) of sub-a-fields

of Z. Then (f / a c A) is called a net asymptotic martin-

gale, or simply a net amart, if

sup {If adv(x*) : x* 1 P} <

holds for every a E A and every continuous semi-norm P, and

limit f du exists in V, where T is the collection of all
TT f L

finite stopping times T : S -+ A, which take only finitely

many values in A and (T = a) e Z for all a E A. The exist-

ence of this limit means that there is an element v in V

such that for each E > 0 and every continuous semi-norm P,

there is a a E T which depends on e and P such that

sup { x*( f d-v) : x* < P} < E

for each T ; T, T 2 0.











Definition 2.3-2. Let (f Z a A) be a net amart.

Then (f Z 2' a A) is said to be uniformly convergent if

for any > 0 there is an element o c T, which depends on

E only, such that for each T ; o, T T,

sup {Ix*( f Td v) : x* < P} <

for every continuous semi-norm P.


Lemma 2.3-3. Let A be a directed set and (f a A)

be a net of functions defined from S to R. If (f a E A)

is Cauchy in U-measure, that is,

limit (f f ) = 0
a ,a' A

in u-measure, then there is a subsequence (fi i N) of

(f a A), and a function f, such that (f i N) con-
1
verges to f u-uniformly.


Proof. Let P be an arbitrary continuous semi-norm

on V. Since limit (f -f ,) = 0 in p-measure, there is
a,a'A a

an element al in A such that

1 1
l lp (s: If (s)-f (s) > ) <

for a, a' in A and a, a' 2 a1. Choose a2 in A such that

a2 > al and

Ilp (5 : f (s)-f ,(s) > )<
p a a a 2 2

for a, a' in A and a, a' 2 a2. Therefore,
1 1
1 1 (s: If (s)-f (s)I < ) <
p a2 2 2











Continuing this process, we obtain a sequence (ai, i s N)

such that
1 1
!Ip (S: fa a(s)-fa l > ) < 2k

for -, a' in A; a, a' a k' and

p (s: If (s)-f (s) I < k
k+l k 2 2

for each k a N. Define

E. = (s: f (s)-f (S) >
a i i+l 2i

then (E )< for each i E N. Define
p i 2i
1o .
Fk = L E. for each k E N, then Iu [i (Fk) k-l
k i=k p 2k-1

We claim that (f a, i s N) is uniformly Cauchy on S Fk'
1
Let e > 0 be given. Then there is an integer m such that
1
S> --- and m > k. Thus
m-l

1
f (s)-f (s) I f (s)-f (S) I < < E
i j n=m n an+l 2-


for each s S Fk and i,j 2 m. This proves that (f) is


uniformly Cauchy on S Fk. Define F = F\ Fk. Then
k=l

F = limit Fk because (Fk, k E N) is a decreasing sequence.
k
Therefore, |4Y p(F) = limit jli]p (Fk) = 0. Assume that
P Pk

s / F, then s V Fk for some integer k. Therefore,

(f (s), i N) is a Cauchy sequence in R. And there is
1
a real-value a such that (f (s) i E N) converges to a .
s a. s
1













We define a function f, f : S -> R, such that

0) if se F
f(s) =
limit f (s) if s E S-F.
1 Cl

Consequently, (f (s), i N) is a Cauchy sequence which

converges to f uniformly on each of the sets S Fk. This

proves that (f., i E N) converges to f i-uniformly.
1

Lemma 2.3-4. Let (f a c A) be a net of p-measurable

functions. If (f a E A) is Cauchy in u-measure, then there

is a subsequence (f i E N) of (f a A), and a p-measur-
ai
1
able function f, such that (f i E N) converges to f
ai
1
p-uniformly.


Proof. From the result of the Lemma 2.3-3 we have

a subsequence (f., i N) of (f a A), and a function f,
1
such that (f i E N) converges to f i-uniformly. We claim
i
that f is u-measurable. Since (f ) is a sequence of i-measur-
ai
able functions for each i in N, there is a sequence of
m m
u-simple functions (g m N) such that (g m N) converges

to f p-a.e. Hence (g m E N) converges to f -uniformly
1 1
from the Theorem 2.1-24. Let P be a continuous semi-norm.

Then there is a set Ak in Z and an integer mk E N such that
1 mk 1
IpAk) <- k and Igk () f (S)
for2 eachk 2s S
for each s E S-Ak*










mk
Define hk = k for each k, then (hk, k E N) is a sequence

of p-simple functions. Define

B =U Ak B = n B, then lp]p (B) = 0.
m k=m m=l
Since (f i N) converges to f p-uniformly, (f ) con-

verges to f p-a.e. Therefore, there is a set A such that

pu p(A) = 0, and (f i) converges to f on S-A. Thus

Iulp (AUB) = 0. We claim that (hk, k e N) converges to

f u-a.e. Assume that s V AUB, then s V A and s V B.

Therefore s V B for some m e N and hence s V Ak for each

k 2 m. Thus

Ihk(s)-f (s) <
k 2

for each k 2 m. Since s V A, (fi (s)) converges to f(s).

Let > 0 be given; then there exists an element at such that
1 E
t 2 m, t c N, < and
2

f (s)-f(s) < F

for each i t, i E N. Therefore, for each i > t we have
1 +
h. (s)-f(s)I h.(s)-f (s) + f (s)-f(s)i < 1 + < .
i 1 a ai2i 2

This shows that (hk, k E N) converges p-a.e. to f and f is

a p-measurable function.


Definition 2.3-5. Let (f a A) be a net of p-measur-

able functions. Then (f a e A) is said to converge to a

u-measurable function f in p-measure if for each E > 0,

limit fIp ({s:SES, If (s)-f(s) > E}) = 0,
for every continuous semi-norm P.
for every continuous semi-norm P.











Lemma 2.3-6. Let (f a E A) be a net of p-measurable

functions defined on S to R. Then (f a A) is Cauchy in

u-measure if and only if (f a a A) converges to a p-measur-

able function f in i-measure.


Proof. It is obvious that the "if" part is true. Now

we want to show that the "only if" part is true also.

Since (f a E A) is Cauchy in u-measure, there exists

a subsequence (fi i N) of (f a. A), and a p-measurable

function f such that (f i E N) converges to f in p-measure.

Let P be a continuous semi-norm on V and let e > 0 be given.

There is a number t such that < E,
2t

p(s: If (s)-f (s) > 1 < 1
Ip at 2t+l t+l'

and
1 1
II p (s: f (s)-f(s) > ) <
p a 2t+l 2t+l
for each a 2 at. Therefore,

I| p (s: f (s)-f(s) > E) I Illp (s: f (s)-f(s) > 2
p a p a, 2 t

l ll p(s: If (s)-f (s) I > 1 )
p a at 2t+l

1
+ I11l (s: If (s)-f(s) >
p at 2t+

< < for each a E A, a 2 at.


This proves that (f a A) converges to f in p-measure.












The proof of the next convergence theorem is modeled

after a similar theorem in Edgar and Sucheston [7, p. 206]

for the real-valued case.


Theorem 2.3-7. Let (f Z a E A) be a net amart.

If (f Z, a E A) is uniformly convergent and

sup { If dv(x*p) : x* < P, a E A} <

for every continuous semi-norm P, then (f a, a E A)

converges to a p-measurable function f in p-measure if

V is sequentially complete.


Proof. From Lemma 2.3-5, we need only to show that

(f a E A) is Cauchy in p-measure. Assume that

(f Z a E A) is not Cauchy in p-measure. Then there is

some E > 0 and some continuous semi-norm P and some
o o
6 > 0 such that for each a E A there is an element
0
a' c A such that a' > a and


llp ( ({s: f (s)-f (s)| > E }) 2 6 .


Choose al arbitrarily in A and let a2 be an element in

A such that a2 1 a1 and

S ({s : If (s)-f 1 ( ) > E 1) 2 6


Because limit f di converges uniformly in the sense that

there is an element v in V such that for each n > 0, there

is a finite stopping time a in T, such that for each T E T,

T 2 0 ,












sup { x*( f dw-v) : x* < P} < n

holds for every continuous semi-norm P. Therefore, there is

some o E T such that
0

sup {}x*(jf d-v) : x* < P} <

holds for every continuous semi-norm P and each T 2 > ,

T E T. Choose a3 iA such that a3 is an upper bound of

a and a3 2 a2. Continuing this process, we obtain a

sequence (an, n N) such that if n is even, an is the

one satisfying an 2 an-l and


U|I P ({s: Ifa (S)-f l (s) > E }) ;
o n n-i

if n is odd, then an is the one satisfying a 2 a n- and

sup { x* ( f dy-v) : x* < P} <
T n

for every continuous semi-norm P and each T 2 an, T s T.

It is obvious that (f n N) is a sequence of p-measurable

functions adapted to (Z n e N). We assert that

(f Z n N) is an L -bounded amart. Since
n n

sup { If dv(x*u): X* < P, a E A} <
f a

for every continuous semi-norm P, then

sup { Ifa dv(x*u): x* P, n N} <
-' n

for every continuous semi-norm P.

Let > 0 be given. Then there is some a E A such
0
"o

that n is even, > and for each T E T, T > a
o n n
O o












sup { x*( f dp-v) :x* < P} < < E


for every continuous semi-norm P. Therefore (f Z ,n N)
n n

is an L -bounded amart. Therefore (f Z n N) con-
n n

verges to a u-integrable function p-a.e. and hence

(f Z n E N) converges in p-measure. But for every
n n
n E N we can always find some integer m, m > n such that


1|1 || ({s: f a (s)-fa (S) I > e ) 0 0o;
Po m m+1

this implies that (f Z n e N) is not Cauchy in p-measure.
n n

Therefore, (f Z a c A) is Cauchy in p-measure and hence

(f a a c A) converges to a i-measurable function f in

u-measure from Lemma 2.3-5.
















CHAPTER III


VECTOR-VALUED ASYMPTOTIC MARTINGALES INTEGABLE
WITH RESPECT TO A VECTOR MEASURE



In [ 2 ] Bartle developed a theory of integration in

which both the function to be integrated and the measure

take values in a relatively general vector space. He con-

sidered there to be a continuous bilinear "multiplication"

defined on the product of the vector spaces in which the

function and the measure take their values, the product lying

in a (possibly different) vector space.

In this chapter we shall first investigate a theory of

martingales which take values in a Banach space, or B-space,

and are integrable with respect to a measure whose values are

in a (possibly different) Banach space. The integrals take

values in a (possibly different) Banach space. Next we extend

these properties to the amart case. The integrability here is

in the sense of the integral defined in Bartle [ 2 ].



1. Basic Concepts and Notations


Let X and Y be real or complex normed vector spaces.

Assume that there is a bilinear mapping, which is denoted by

juxtaposition, defined on X x Y with values in a Banach space

Z, satisfying xy l K Ix I y for some fixed positive number K.

44










Let S be an abstract set and Z be a field of subsets

of S. Let p be a finitely additive set function defined

on Z to Y. The semi-variation of p is the extended non-

negative function |luI whose value on a set E in Z,

denoted by IEI or IHpl (E), is defined to be

l|pII (E) = sup Zxi (Ei) ,

where the supremum is taken over all partitions of E into

a finite number of disjoint subsets (Ei) C and all finite

collections of elements (xi)CX with 1xi 1. The

variation of u is the extended non-negative function [p ,

is defined by

|o| = sup ZEu(Ei)

where the supremum is taken over all partitions of E into

a finite number of disjoint sets (Ei)CZ.

It is obvious that the semi-variation of p is a monotone,

subadditive function on Z, and that the variation of i is

a monotone, additive function on Z. Also, if E is in Z, then

0 IuIll (E) 5 K- I (E) < .

We extend the definition of I1pll and Ipl to arbitrary

subsets of S as follows: if ACS then

I[Al = inf {IfEI: E E 2, AC E}
and

I[ (A) = inf { p (E): E Z, ACE}.

It is evident that the extension of I|ill agrees with its

former value on Z and is a monotone, subadditive function

on the collection of all subsets of S.












A subset E of S is a p-null set if liE11 = 0.

A proposition holds p-almost everywhere, abbreviated as

u-a.e., if it holds outside of a i-null set. A p-simple

function is a function f: S -> X which takes only finite

many values x ,... ,x each non-zero value x.. being taken

on a set E. in Z with |IEil < m. Thus f can be represented

as
n
f = Z x I E. E Z,
i=l 1 i

where I is the characteristic function of the set E..
1
We define the integral of a simple function

n
f = Z x. I, E.E E over a set E in E by
i=l1 1

n
E fd = x. (E0E ).
i=l 1

It is easy to see that the integral of a simple function is

independent of the representation given in its definition.


Lemma 3.1-1 [ 2 p. 340]. Let w be a finitely aditive

set function from Z to Y.

(a) For each fixed E in Z the integral over E is a

linear mapping defined on the linear space of

simple functions on S to X, and has values in Z.

(b) For each fixed simple function, the integral is

a finitely additive set function on E.











(c) If f is a simple function and f(s) < M for all

s in E E Z, then

I fd M IEl .

A sequence (f ) of functions on S to X is said to

converge in p-measure to a function f if

I {SeS: Ifn(s)-f(s) I E }I -- 0

as n -- m for each E > 0.

A function f: S ->- X is p-measurable if it is the

limit in measure of a sequence of simple functions. Clearly,

the collection of all measurable functions on S to X is a

linear space which is closed under the operation of conver-

gence in measure of sequences. A sequence (f ) of functions

converges p-almost uniformly to a function f on S if for

every e > 0 there is a subset A of S such that I|A El < E

and (f ) converges to f uniformly on S-A .


Lemma 3.1-2 [2 p. 340]. Let p be a finitely additive

set function on Z to Y.

(a) p-almost uniform convergence implies convergence

in u-measure to the same function.

(b) u-almost uniform convergence implies u-almost

everywhere convergence to the same function.

We shall assume that p is a finitely additive set func-

tion from S to Y from Definition 3.1-3 through Theorem 3.1-10.











Definition 3.1-3 [2 p. 341]. A function f, f: S->-X,

is said to be i-integrable over S if there is a sequence (fn)

of u-simple functions on S to X such that the following

conditions are true.

(1) The sequence (fn) converges in p-measure to f.

(2) The sequence (X ) of indefinite integrals

i (E) E- ffdp, E E

has the property that given any S > 0 there is

a 6 > 0 such that if E is in Z and I|El|< 6, then

| n (E)< E, n = 1,2,... .

(3) The sequence (X ) has the property that given any

E > 0 there is a set E in Z with IE I < m and

such that if F is in E and FCS E then

SnI (F) < s, n = 1,2,... .

Condition (2) is frequently stated as "the sequence

(n ) is uniformly absolutely continuous with respect to

|11l ," and (3) is stated as "(A ) is equicontinuous with

respect to ul |I."


Theorem 3.14 [2 p. 341]. If f is integrable over S

in the sense of Definition 3.1-3, then for each E in Z

limit A (E) exists in the norm of Z. This limit is denoted
n-nco
by X(E) or by [Efdp and is called the value of the indefin-

ite integral A at E, or the integral of f over the set E.

In addition, the limit A(E) = limit A (E) exists in the norm
n
of Z uniformly for E in Z.











The integral is independent of the sequence of simple

functions used to define it.


Theorem 3.1-5 [ 2 p. 342]. (a) If E is in Z then the

set of functions integrable over E is a linear space and the

integral over E is a linear mapping of this space into Z.

(b) If f is integrable over S, the integral of f is

a finitely additive set function on the field E.

(c) If f is integrable over S, then

limit Efd = 0.


(d) If f is integrable over S, then given any E > 0

there is a set E in Z such that if F E Z and

FC S-E then

|Ffdu < E.


Definition 3.1-6 [ 2, p.342]. A function f, f: S -+ X,

is w-essentially bounded on a subset A of S if

inf sup f (s) < c,
N seA-N

where the infimum is taken over all u-null sets N. We write

ess sup jf(s) for this number.
sEA

Theorem 3.1-7 [2 p. 342]. If f is an essentially

bounded measurable function then f is integrable over any

set E in Z with EII < m and


IEfdiu {ess sup f(s) } [Ei.
sEE











Theorem 3.1-8. (VITALI CONVERGENCE THEOREM) [ 2, p. 343].

Let f be a function on S to X and let (f ) be a sequence

of integrable functions such that

(1) the sequence (fn) converges in p-measure to f;

(2) the sequence of indefinite integrals is uniformly

absolutely continuous with respect to |III ;

(3) the indefinite integrals are equicontinuous with

respect to I P .

Then f is an integrable function and

TEfdu = limit fEf dw, E E.
n n

Furthermore, the limit is uniform for E in E.

In the case when either X or Y is a scalar normed vector

space, the integrability of a function f implies that of the

function f(-) ; but this is not true here in general, as can

be seen in the following example.


Example (a) [2 p. 344]. Let X = Y = real Euclidean

three-space, with the usual inner product, and let 61, 62 and

53 be the unit coordinate vectors. Let X be Lebesgue measure
-1
on S = [0,1] and let p(E) = A(E)61, g(s) E s1 2. Then g

is u-integrable; but Ig() is not integrable.

Also the usual formulation of the Lebesgue Dominated

Convergence Theorem is not true here in general, as can be

seen in the following example.











Example (b) [2 p. 3441. Let S, X, Y, p and g be as

in Example (a). If f (s) s1(l/n) 61, then (f ) is a

sequence of integrable functions such that Ifn(s) I g(s)
-1
and (f ) converges p-a.e. and in p-measure to f (s) s 65

But f is not integrable.

The converse of the Vitali Theorem fails in general.

This can be shown in the following example.


Example (c) [ 2, p. 344]. Let S, X, Y, and v be given

as in Example (a). Let h (s) 1 62 and ho(s) E 1 63'

Then each h is integrable and


{E hod = limit E hndp = 0
n
uniformly for E E Z. Conditions (2) and (3) of the Vitali

Convergence Theoem are satisfied; but (h ) does not converge

to h at any point and not in measure.


Theorem 3.1-8 [2 p. 345] (DOMINATED CONVERGENCE THEOREM).

Let (f ) be a sequence of integrable functions on S to X

which converges in measure to a function f. If there is an

integer g such that for each E in Z and n E N, where N is

the set of all natural numbers, we have

I'E fndyI iE gd j ,

then f is integrable on S and

limit E fnd = E fdp for each E Z.
n











Theorem 3.1-9 [2 p.345]. Let (f ) be a sequence of

p-integrable functions from S to X such that (f ) converges

to a function f in measure. If fn(s) M for almost all

s E S, then f is integrable over any set E in Z with IIEll <

and

limit fE ndY = E fd.



Theorem 3.1-10 [2 p. 345]. Let (fn) be a sequence of

u-integrable functions which converge almost uniformly to f.

Then f is integrable over any set E in Z with I|El < and

limit fE fnd = jE fdy.
n
From now on we shall assume that U is a countably addi-

tive set function from o-field Z to Y. Under this assumption

we can prove that if f: S -+ X is p-integrable, then the

indefinite integral of f is a countably additive set func-

tion on Z to Z.


Definition 3.1-11 [2 p. 346]. Let p be a countably

additive set function on a o-field Z to Y, then p has the

*-property (with respect to X) if there is a non-negative

finite-valued countably additive measure v on Z such that

v(E) -- 0 if and only if |KI 0.

One can see that if 1 has the *-property, then

IIS I< and li|p| is countably subadditive on subsets of S.











Theorem 3.1-12 [2 p. 346]. Assume that the countably

additive set function Uhas the *-property, then the following

are true.

(a) If a sequence (f ) of functions on S to X converges

in p-measure to a function f, then some subsequence

converges p-almost uniformly to f.

(b) If a sequence (f ) converges p-almost everywhere

to f, then it converges p-almost uniformly to f.


Theorem 3.1-13 [ 2 p. 347]. Assume that the countably

additive set function p has the *-property. Then a function

f on S to X is integrable if and only if there is a sequence

(fn) of simple functions such that
(1) the sequence (f ) converges to f almost everywhere;

(2) the sequence (X ) of indefinite integrals converges

in the norm of Z for each E in Z.


Theorem 3.1-14 [2 p. 347]. Assume that the countably

additive set function p has the *-property. Let (f ) be

a sequence of integrable functions such that

(1) the sequence (f ) converges p-a.e. to f;

(2) given E > 0 there is a 6 > 0 such that is E in E

and |IE < 6 then

If %du < n n E N.

Then f is integrable on S and

E fdp = limit E fndp,
n
uniformly for E E Z.












2. Martingales


Throughout this section we shall assume that (S,Z) is

a measurable space; that is, S is a set and Z is a a-field

of subsets of S; X, Y, Z are Banach spaces, p: Z -> Z is

a countably additive set function, and (En, n N) is a
00
sequence of sub-a-fields of Z with Z = a( U n ).
n=l

Definition 3.2-1. Let (f ) be a sequence of p-measurable

functions from S to X adapted to a sequence of sub-a-fields

(n n N). We say that (f n, n N) is a martingale if

and only if the following are true.

(1) For each n E N, f is a p-integrable function.

(2) For every fixed n, n E N,

f du = f
IE n = IE fm

for each E Z and each m e N, m n.
n
A -measurable function T, T: S -- N, is said to be

a stopping time for (Z ) if (T=n) En for each n N.


Definition 3.2-2. A class W of subsets of S is said

to be a H-system if it is closed under the formation of

finite intersections:

(I) A, B E W implies AAB E W.

A class L is a A-system if it contains S and is closed under

the formation of proper differences and countable, increas-

ing unions:












(1) S L ;

(X2) A, B E L and ACB imply B-A EL;

(X ) A,A2,. .. L and An t A imply A E L

We remark that a class that is both a H-system and a

X-system is a o-field.


Theorem 3.2-3. (Dynkin's 1i-X Theorem) [ 3, p. 34].

If W is a H-system and L is a X-system, then W< L implies

that o(W) C L.


Lemma 3.2-4. Let p be a countably additive set function

on Z to Y and assume that there is a non-negative finite-

valued countably additive measure v on Z such that

|ulI << v; that is, for each E in Z v(E) --> 0 implies

|i (E) -> 0. Then each B E o( U n) = Z and for each
CO n=l
S> 0 there is a set A E U such that ||w (A A B) < e.
n=l

Proof. Let L be the collection of all sets in Z

satisfying the following property:

If E lies in L, then for each E > 0, there is a set

E in U Z such that v(E A E ) < E.
n=l n

It is clear that U Z C L and U Z is a H-system.
n n
n=l n=l
We claim that L is a A-system.
00
(1) S is in L because S is in U E
n=l

(2) Assume that A, B are in L and ACB.

Let E > 0 be given. There are A BE in U Zn such that
n=l












v(BAB ) < and v(AAA ) < -


Since (B-A) A (B -A ) C (BAB ) U (AAA),


v((B-A)A(B -A ) \)((BAB ) U (AAA) )


< v(BAB ) + V(AAA ) < F.
00 C
Also B -A U E because U Z is a field.
S n= n n=l

Therefore B-A is in L.

(3) Assume that A ,A2,..., are in L and An A.

Let B = A-A then B + 4. Since v is a countably additive
n n n
set function on Z we have v(B ) + 0.

Let E > 0 be given. There is an integer n such that

\(B )< -. Since A E L there is a set
n 2 n
o o

F E UJ Z such that v(Fn A A ) <
n n 0 n 2
o n=l o o


Now, A A F = (A-F ) U (Fn -A)C (Bn U (An -F ))
n n n n n n
0 0 0 0 0 0

U (Fn -A ) = B (An AF ).
0 0 0 0 0

Therefore, v(A A F ) n U(B ) + (A A F ) < C.
o o o o

This implies A E L. By Dynkin's H-X theorem we have


0( U 2 ) C_ Therefore L = o( LV Z ).
n=l n=l

Since |I|fi 0 there is a 6 > 0 such

that v(E) < 6 implies \\i (E) < E. Therefore for each B
00
in (( U Z ) and for each E > 0, there is a set A in
n=l n











U En such that v(D A A) < 6 and hence |iAA l| < e.
n=l

This completes the proof of the lemma.


Theorem 3.2-5. Let u be a countably additive set

function on Z to Y and v be a non-negative finite-valued

measure on Z such that |i-i| << v. Assume that

(fn, n n e N) is a martingale. If there is a

p-integrable function g such that

IE fn du I TE gd l,

for each E in Z and each n E N, then

limit f dp exists for each E in a( U E).
n fE n n-1 n
n n=l

Proof. First assume that E is in U ; then E is
n=l n
in Z for some integer n From the definitionof martingale
0
we have

fE f = E fndu, for each n c N, n 2 n .



CO
Therefore, limit Ef ndU = Efn du for each E E n
n o o

Next we assume that E E Z but E U E Let E > 0
n=l
be given. Since g is o-integrable, limit gdp = 0,
101-0 B

there is a 6 > 0 such that ill < 6 implies IB gdu < 4.

From lemma 3.2-4 there is a set A in J E such that
n=l
|FA [I < 6. Ac is in n for some integer n ; therefore

f du = f md for n, m E N, n, m n .
A n fA o 0










Now let n, m be integers with n, m 2 n ;then

T f du Ef dU i f dU Af dpi
E n E E

+ f n dpi- ff MdU = | f dp ( f dp +
+ n d Em E n -A fn



IA f mdp-Ef dp .
A E


Also,

Ef dp f dp[ = i f d f dp
E (EnA )U(E-A ) (EnA )U (AE-E)

= fndp f f nd nd + f fndpI
E-A A -E E-A A -E
E E C

SE-gdl + I gdp .
E-A A -E


Since Ivi 1 is a monotone function on E,

p|i (E-A ) p II (EAA) < 5 .

Therefore if gdl~ < .
E-A
F-


This implies that

EndU iA fnd 1<


Similarly, IE fmdu fmdp[
E

Consequently, f dp f di < e
E N.
n, m no, n, m e N.


c-
1.
2"

< -
2for each

for each


This proves that {E f dp, n E N} is a Cauchy sequence

in Z. Therefore, limit i f dl exists, because Z is
n n


a Banach space.











Lemma 3.2-6. Let T be the collection of all bounded

stopping times for (Z n N) and (fn n E N) be a sequence

of v-integrable functions adapted to a sequence (Z n N).

Then (fn Z, n e N) is a martingale if and only if f du

does not depend on T, T T.

Proof. Assume that (fn, n, n E N) is a martingale.

Let T T2 be arbitrary stopping times in T,

max T (s) = ml, max T2(s) = m2 and mI 2 m2'
sES sES
Then

df du f dp = S fk du
~1 m 1T k=l (T =k) k
U Tl lk) 1
k=l
mi
Z= f dw = f dp.
k=l (T1=k) 1 1
m2
fd = m2 f du = Z fkdi
2 U (T2=k) 2 k=l (T2=k)

m2
k=l
2
= Z f du = f du.
k=l (T2=k) m

Therefore

fI du = f 2di for each T1, T2 E T.


This proves that f dU does not depend on T, T c T.

Conversely, assume that f Tdu doesnotdepend on T E T.

Let n be an arbitrary but fixed integer and B be a set in ZEn

For each m E N, m 2 n define two functions T1, T2on S to N

as following:











n if s E B
T1 (S ) c
m+l if s E B

2 m if s B
T2(S) '
m+l if s E B .

Since (Tl=n) = B n and (T--m+l) = B c ZC m+l, Zi is

a bounded stopping time for (Zn' n c N).

Similarly T2 is a bounded stopping time for (Zn)

f du = fT d + fTd = n f d + f fm+d
B B


and f f d = fdu + i fT d = f md + B f +dp.
2 )B2c m2 cm+l


But f Tdu = fT du, hence f dp = f fmd1
j L 1 2 fB nB

This proves that (f Z n E N) is a martingale.

Lemma 3.2-7. Let (f Z n N) be a martingale and

k be an arbitrary but fixed integer. Let A be a set in Zk'

then I f du = Afkdp for each T 2 k, T E T.
A A


Proof. Let T be a bounded stopping time in T and

max T (s) = m. Define a function T1: S -+ N by
sS
) ~ T (s) if S A
T1(si) =
if s A
m+l if s A .











For each n N, k < n < m, (T =n) = (T=n) /) A Zn

and (T -m+l) = Ac E Zk C E+1; therefore T1 is a bounded

stopping time for (Zn)

Define T : S -> N by

k for s E A
T2(s) =
m+l for s E A.


It is obvious that T2 is in T. Since

fA d f du -f dui J Acf mdu
A

fT du + f du= f du fT dp = 0.
) 2 J c 1 T 2

Therefore,

Af du = A fkd for each T k, T E T.
A ^A

Theorem 3.2-8. Let u be a countably additive set function

on Z to Y and v a non-negative finite-valued measure on Z

such that |lu| << v. If (f n' En n E N) is a martingale

and there is a u-integrable function g such that

|'Bf Tdu I gdp ,

for each T c T and each B c a( VJ E ), then
n=l

limit f du exists for each B E a( Z ).
TET -B n=l
00
Proof. First assume that B is a set in EZ. Then
n=l
B is in E for some n N. From Lemma 3.2-7 we have
n o

f dp = fn du for each T E T, T 2 n
'B T o












Therefore

limit fBfT d = Bf dp.
TET B B o


Next assume that B Z and B i U Z By the same argument
n=l
as in Theorem 3.2-5 and the fact that

If Td = Bfkdu for each BE k Z
B f B

for each T c T, T k, we can show that

limit f dI exists for each B .
TET nB n=l

Theorem 3.2-9. (Vitali-Hahn-Saks) [6, p. 158].

Let (S, Z, A) be a measure space, and (XA) a sequence of

X-continuous vector or scalar-valued additive set functions

on E. If the limit, limit A (E), exists for each E in Z then
n
limit 1 (E) = 0, uniformly for n = 1,2,....
v(X,E)0 n

In addition, if v(X,S) < -, the function

F(E) = limit A (E) is countably additive on Z.
n

Theorem 3.2-10. Assume that P is a countably additive

set function on Z to Y and that p has the *-property. Let

(fn, n, n E N) be a martingale and g be a p-integrable
function such that

f du gdU ,
{E fn Igdl'E

for each E E Z and each n c N.

Let X (E) = f du for each E E Z and define F(E) = limit A (E)
it is well defined from Theorem 3.2-5)
(it is well defined from Theorem 3.2-5).











Then F is a countably additive set function on Z to Z and

F << p I; that is, for E E IZ |--- 0 implies F(E) -- 0.


Proof. Since f is -integrable,

limit f nd = 0; that is, limit A (E) = 0.
iEE I0 E In I 0 n

By the hypothesis, v has the *-property, and there is a

non-negative, finite-valued set function von Z such that

|E[|-> 0 if and only if v(E) -- 0. Therefore

limit A (E) = 0 for each E c Z.
v(E)0 n

Also

limit A (E) = F(E) exists for each E E Z.
n-o n
From the Vitali-Hahn-Saks theorem we have that F is a

countably additive set function on Z to Z, and

limit A (E) = 0 uniformly in n c N.
v(E)0 n

Let E > 0 be given. There is a number 6 > 0 such that

for each E E Z with (E) < 6 implies

S (E) <- n = 1,2,3,....
2
Let E 0 E and v(E ) < 6. Since limit A (E ) = F(E ),
n
there is an integer no such that

I (E ) F(Eo) < ,


and hence F(E ) < IX (E )I + E < This proves that there
o
is a number 6 > 0 such that for each E in Z with v(E)< 6

then IF(E) < Therefore F << v. But v << I\II and

hence F << II Ii .












Corollary 3.2-11. Under the hypotheses of Theorem 3.2-10

and the additional hypotheses that G(E) = IEgd1 for each

E in Z and G has finite variation, F has finite variation.


Proof. From Theorem 3.2-10 we have that F is a countably

additive set function on E to Z. Let E be in E,
n
v(F,E) = sup E JF(E )
i=l 1

where the supremum is taken over all partitions of E into

a finite number of disjoint sets in Z. Let {E,,...,En}

be an arbitrary but fixed finite partition of E; then

F(E) = limit f dp.
n E.
1
There is an integer m. such that
1

) f dw F(Ei ) <-
m. i n
E. l
1
and hence

F(E) < f d1i + 1 l gdvl + 1
i mi n n
i 1i

for each i = 1,2,...,n. Therefore,
n n
E IF(E.) < E gdi [ + 1 E v(G,E) + 1.
i=l i=l E

This implies that v(F,E) < v(G,E) + 1 < m. This proves

that F has finite variation.

We shall extend the next two properties from the similar

properties of the real-valued random variable case which were

proved by Austin, Edgar and Tulcea in [1, p. 18].












Lemma 3.2-12. Let (fn, n N) be a sequence of

X-valued, p-measurable functions. Let Zn = o(f ,f2," ,fn)

for each integer n 2 1 and assume that

Z = a(J Z ) .
n=l
Let h be an X-valued, p-measurable function such that for

each s E S, h(s) is a cluster value of the sequence

(fn(s), n E N).

If u has the *-property, then given any E > 0, 6 > 0,

and integer m 1, there is a bounded stopping time T for

(z n E N) such that T m and

1lp1 ( {s: If (s)-h(s) 2 6} ) < .
S(s)

Proof. By the hypothesis h is a p-measurable function,

there is a sequence of p-simple functions (h n N) such

that h -n- h in p-measure. Let c > 0, 6 > 0, and integer

m 2 1 be given. There is a simple function h such that
6 E
|| || (s: hn (s)-h(s)| 2 ) < q.
o


Define




and let


E = {s: |h (s)-h(s) > -|}
n 2
mo
h = E x IE
o k=l k


mo
where U Ek = E, Ek Z and xk X. From Lemma 3.2-4
k=l
we have that there are sets A1,A ,...,A in U Z such that
1 2 m n
o n=l

[III (EkAAk) < and Ak Z for each k = 1,2,...,mo.
ko 4m k










mo mo
Define g = E xk (EkAAk) D,
o k=l k k=l

and M = max {m, nl,n2,...,n }. Then gn is M-measurable.
o o
We claim that
mo
{s: gn (s)-h(s) } C E ( j (Ak-Ek
o k=l


Assume that s E {s: g (s)-h(s) and s E.
o n 2
0

Now if s is in Ek n Ak for some k, lk<:mo, then

h (s ) = gn(s) and hence gn (so)-h(so) <
o o o 2

This contradicts s E {s: g (s)-h(s) 2I
o n 2
0
mo
Therefore, so U (Ek nAk )
k=l
mo mo
But J (Ek Ak) ( ( U (Ak-Ek)) = S;
k=l k=l
mo
then s C U (Ak-Ek)
k=l
This proves the assertion. Therefore,




0



c h 4 o 4m 20
k=l o

Since p has the *-property, |)H < m. Therefore

|| ({s: (s)-h(s) < -h(s) > })
o
o







67



Since h(s) is a cluster value of the sequence (f (s) ",n N),

we have
6
{s: |h(s)-g (s) < } C {s: f ()-gn (s) I<
0 0


for some n 2 M} = U {s: If (s)-gn (s)l < } Thus
n=M o
6 6
v({s: h(s)-gn (s) < -) v( U {s: f (s)-gn (s) o n=M o
k 6
Define Bk = {s: f (s)-gn (s) <-} for each k e N, k > M,
n=M o

and B = U {s: fn(s)-gn (s) < 3. Then {Bk, k > M}
n=M o
is an increasing sequence converging to B and hence

v(B-Bk) -- 0 as k -* m,

and Ipll (B) L |IiPI ({s: |h(s)-gn (s) < ) > 1- .
O

Assume that |I I (Bk) 11- for each k e N, k > M.

Define a = sup ||p| (Bk) ;then a < is|| -
k>M

Let = I -a; then E > 0. Since p has the
o 2 o

*-property, there is a non-negative finite-valued measure

v on Z such that iuil << v and hence there is a number

6 > 0 such that v(G) < 6, G Z implies IIu (G) < .

Since limit B-B = k there is an integer k such that
k
v(B-B ) < 6. Therefore pIl I (B-Bk )< o,
o o
p | (B) < l | (Bk ) + ii (B-Bk ) < a + = .
0 0


l jj1 (B) > |S I| .
-25


This contradicts










Therefore fipl| (BN,) > |S|| for some N' E N and N' > M.

Define a function T : S --N by

T(s) = the first n such that M < n < N'

and If (s) g (s) i -
o
= N' otherwise.

Since g is EM-measurable, if n E N, M o
(T=n) = {sES: Ifk(s)-g (s) > 5 for each integer
o
k, M < k < n} C Z n. If n = M, then

(T=M) = {s: IfM(s)-g (s) } Z M. If n = N', then
O
(T=N') = {s: fN' (s)-gn (s) } U {s: f (s)-gn (s) > ,
o o
for each n, M < n N'} c EN,. This implies that T is a

bounded stopping time for (En, n N). Therefore,

;I|p ({s: If ) (s)-gn (s) } > |s|| -J

and hence II ({s: If(f (s)- (s) > }) <
O
Now {s: f (s)(s)-h(s) 2 6}C {s: If (s) (s)-gn (s)I >
O
C {s: (g (s)-h(s) 2}.
o
Therefore

[ 1 ( ({s: f (s) (s)-h (s) 6})

< | ({s: f (s)-gn (s) > 1)

+ | ({s: gn (s)-h(s)| 2 3})< .
o


This completes the proof of the lemma.











Theorem 3.2-13. Assume that p has the *-property.

Let (f n N) be a sequence of X-valued, u-measurable

functions. Let Zn = o(f ,f2".. ,fn) for each n N and

o( z E ) = n If h: S -- X is a p-measurable function
n=l
such that for each s E S, h(s) is a cluster value of the

sequence (f (s) n N), then the following are true.

(1) There is a strictly increasing sequence ( n,nN)

of bounded stopping times for (Z ,neN) such that

limit f (s) = h(s), p-a.e.
n n
(2) If (f ) is a sequence of p-integrable functions,

then h is u-integrable and

limit f du = jhdu for each E E.
n E n E


Proof. (1) Let E > 0, 6 > 0, m z 1 be given. From

Lemma 3.2-12 there is a bounded stopping time T for (En,nsN),

T 2 m such that

1I ll ({s: f f (s)( -h(s) I 6) < .

We obtain an increasing sequence of bounded stopping

times (T n' n N) as follows:

When n = 1, we choose T1 T such that T1 2 1 and

ull i ({s: If () (s)-h(s) 1) 5 1;

when n = 2, we choose T2 T such that

T2 L max {T1(s): sS} and

1 1
) )){s: )f (s) (s)-h(s) ;
2












by induction, we have that for every n,

T 2 max {T (s): seS} and
n n-1
1 1
IIPII {s: If (s)-h(s) 2 -} -

We claim that (f n N) converges to h in p-measure.

Let e > 0, 6 > 0 be given. There is an integer no such that

S> 1 and E > -. If n E N, n 2 n then
n n o
o o

||u/ j ({s: f T (s)-h(s) 2 6})
n
1 1
II Hp ({s: If (s)-h(s) ) < < .
T n n n

This proves the assertion. Since p has the *-property from

Theorem 3.1-12, there is a subsequence (Tkn, n a N) such that

(f n E N) converges p-almost uniformly to h, and therefore
Tk
kn
converges p-almost everywhere to h.

(2) Since (f n E N) is a sequence of p-integrable

functions, (f n E N) is a sequence of p-integrable func-
kn
tions which converge p-almost uniformly to h. Since p has

the *-property, lp II (S) < -. Therefore, from Theorem 3.1-10,

h is a p-integrable function over every set E in Z and


limit f dP = Yd.
n E nk E

In [1], Austin, Edgar and Tulcea used the same property

of the preceding theorem to show that a sequence of real-

valued amarts comverges almost everywhere to a function under











some conditions. We cannot extend this convergence pro2mrty

here, because it is not true in general that if two U-inte-

grable functions f and g satisfy fdp = gdy for each
B JB
B in then f = g p-a.e. This can be shown in the following.

Example. Let X = Y = real Euclidean three-space, with

the usual inner product. Let 61, 62 and 63 be the unit

coordiate vectors and A be Lebesque measure S = [0,1] and

w(E) = A(E)61. Let A = [0,1, define two functions f, g

from S to X by

f = IAc 63 + A 61

g = IAc 2 + IA l'

Then for every B in Z we have

B fdu = X(A n B) = B gd
B B

But for each sAc, f(s) # g(s); and IeAC1 =


Theorem 3.2-14. Assume that p has the *-property.

Let (fn' Zn n N) be a martingale, Zn = (fl .f n) for


every n e N and Z = a( U Z ). Let h: S -- X be a
n=l
a u-measurable function such that for each s in S, h(S)

is a cluster value of the sequence (f (s), n N). If there

is a p-integrable function g such that


E f du l E gdf d ,
E d E

for every T c T and each E E Z then

limit fTdu = Ihdu for each E in Z.
TT E E












Proof. From Theorem 3.2-8, we have that limit fT di
TET E

exists for each E in Z. Also from Theorem 3.2-13, we have

that there is an increasing sequence of stopping times

(Tn' n N) such that for each E in Z

limit f Tdp = hdi.
n fE n E

Therefore, limit f di = hdv, E E.
TT E E



3. Asymptotic Martingales


In this section we shall continue to use the same

notation used in Sections 1 and 2.


Definition 3.3-1. Let (f ) be a sequence of p-measurable

functions from S to X adapted to a sequence of sub-a-fields

(Zn, n E N). We say that (f En, n e N) is an asymptotic

martingale, or amart, if and only if the following are true.

(1) For every n in N, f is a p-integrable function.

(2) The limit,

limit f d exists in Z,
T T T
where T is the collection of all bounded stopping

times for (Z n N).

From Lemma 3.2-6, we have that every martingale is

an amart.

Now we shall investigate several properties for vector-

valued amarts which are integrable with respect to vector-

valued measure.











Proposition 3.3-2. Let (fn, E n N) be an amart

and IIS 1| < -. Then

sup {I f dpl: T E T} < .


Proof. First we want to show that if f is a p-integrable

function from S to X then

sup { fd | i : E E L} < .
SE


From

(gn,


There


Assume that f is a p-integrable function on S to X.

Theorem 3.1-4, there is a sequence of p-simple functions

n E N) on S to X such that

I fdp = limit gn d, uniformly for E in E.
E n n
fore, there is an integer N1 such that for every E in Z,

SgN dP fdp| < 1.
^E 1 ^E


Thus,


where


I du< 1 + I g dpI < 1 + M- S

M = max {IgN (s) : S a S} < m
1


from Theorem 3.1-7. This proves the assertion.

Now, (f ) is an amart, and there is an integer no

such that

I f du fn dp < 1, T T, T 2 n .
o


Therefore,


where


fT dui < 1 + Iffn dpj 1 + M

M = sup{ [ f nd : E Z} |S I.
o n







74



If T is a bounded stopping time with T < N2,then
N2 N2 fd
f fdu| = E fkdl fkd~l
k=l (T=k) k=l (T=k)
N2
S E (1 + Mk)
k=l

where Mk = sup {| EfkdU1I: E Z} .

Let K = max {l+Mo, 1+M1, 1+M2, ..., 1+MN2

then sup {I f du : T T T < K < O.


Theorem 3.3-3. Assume that p has the *-property.

Let (f Z n c N) be an amart and Z = o( U E ).
n n n
n=l
If there is a u-integrable function g such that

fE f du < If gdi E c E, T E T,
JE T )E

then for every E in Z limit f IEdu exists.
TET E

Proof. Since g is p-integrable, let E > 0 be given;

then there is a number 6 > 0 such that E in Z and II Ei < 6

implies gd < 6.

Let E E E be given. Since U has the *-property, there

is a set A in some sub-a-field Z such that jIIjI (AAE) < 6
0
from Lemma 3.2-4. Since (f n N) is an amart, there is

an integer nl such that

If du f di < ,' 1,01 T1 2 nl
I aT 1 C G 1 3a11











Define n2 = max (n ,n ). Let T, o be arbitrary bounded

stopping times in T, T, a > n2. Define T1,o1 from S to N by

T (S) = T(s) for s c A

= n2 for s V A

and al(s) = o(s) for s A

= n2 for s A.

Then (T1 = n2) = ((T=n2)(A) U Ac Z


and (T1 = = = (T = m)n A E ,m, m E N, m 2 n2;

because A is in Z This imDlies that T is a bounded
n2 1
stopping time. Similarly, a1 is also a bounded stopping

time. But


f d f a dp + f G d f ad

+ if A d + f di f d


f d f Tdp + f dp1 -f dp

+ f f fdp Af di
E-A A-E

S2 gdu| + 2 gdpl + 5 e.
E-A A-E

This proves that for each E in Z

limit f dp exists.
TT fE T


Theorem 3.3-4. Assume that y has the *-property.

Let (f ) be a sequence of P-integrable functions adapted

to an increasing sequence (Zn, n N) of sub-a-fields of

E. If (fn) converges P-almost everywhere to a P-measurable

function f, then (fn Zn, n E N) is an amart.











Proof. Let (T n E N) be an arbitrary increasing

sequence of bounded stopping times for (Zn, n e N). Since

(f n E N) converges u-almost everywhere to f, (fT n N)
n

converges i-a.e. to f also. Since p has the *-property,

from Theorem 3.1-12, (f n e N) converges p-almost
n
uniformly to f.

From 3.1-10, f is integrable over S and

Sfdp = limit f di,
n n
because u has the *-property, ilsIl < <. This implies that

limit f d = fdw.
TT

Therefore, (f Z n e N) is an amart.


Theorem 3.3-5. (The optional sampling theorem for

amarts.) Let (fn' En, n E N) be an amart and (Tk, k E N)

be a non-decreasing sequence of bounded stopping times for

(Zn, n N). Define g = f and Jk = = {A C :
nk T k TK

A ( ((Tk = n) E Zn for all n}.

Assume that p has the *-property. Then (gk' Jk' k e N)

is an amart.


Proof. We shall complete the proof in the following

three steps.

Step 1. We claim that gk is a Jk-measurable,

p-integrable function for each k N. Let k be an arbitrary

integer in N, m = max {Tk(s) : s S} and

An = (k = n), n = 1,2,...,mo'











then A c Z n = 1,2,...,m For each n, n = 1,2,...,m ,
n n o
m
there is a sequence of p-simple functions (h m e N) such

that (hm, m c N) converges to f in p-measure, and the
n n
indefinite integral of {hn, m N} are uniformly absolutely

continuous and equicontinuous with respect to II11|.

Let 6 > 0, E > 0 be given. There is an integer M such that

if m M, then

lu ({s: hhm(s) f (s) > 6}) < n = 1,2,. ,m
n n m 1,2,,mo
0
o
Define a sequence of simple functions (gk, m e N) on

S to X as follows:

gm(s) = hm(s) if s A n = 1,2,...,m .
k n n o

Since A n (Tk = t) = A if t = n

= if t # n,

An Jk for every n = 1,2,...,m .
n k o

Thus gk is a Jk-measurable function. Also, for each m, m>M,

|lu| ({s: Igm(s) gk(s) > 6

= I ({s: Ig (s) f (s) > 6})

m
= ( U i{s: |hm(s) f (s)I > 6})
n=l
m
o
< !lull({s: |hm(s) f (s) > 6}) < E.
n=l n n

Therefore, (gk, m N) converges to gk in u-measure.











Define X (E) = g dpi for each m E N and E E.
m fE "k

We claim that (m m N) is uniformly absolutely continuous

with respect to II|l| and (X m E N) is equicontinuous with

respect to KI'I.
m
Since the indefinite integrals of (h me N) are

uniformly absolutely continuous with respect to I|p i,given

any E > 0, there is a number 6 > 0 such that F in Z and

IlvI! (F) < 6 implies

h du< -, n = 1,2,...,m m N.
F n m 0
Let E be a set in Jk with lii < 6, then

ll|l (E n n An) < 6, n = 1,2,... ,m .

Therefore,
m
gkdul < hmdpl < E, m E N.
E n=l EA n

This implies that (X m c N) is uniformly absolutely con-

tinuous with respect to I(iPI.

Also the indefinite integrals of {h m e N} are equi-

continuous with respect to |IIIl. For any E > 0, there are

sets En, n = 1,2,...,m in Z such that IEn,< and-if

F C S En, F c Z, then

Ihmdv < --, m E N, n = 1,2,...,m .
Fn m 0

Define E = En; then IE l < m, and if F in Jk,
n=l
F C S E then F c S En, F Z, for every
n = 1,2,...,m and
n = 1,2,...,m and











m
gmd Z | hmdyl < E, m c N.
F n=l FA n'
Sn

This proves that (X m E N) is equicontinuous with respect

to ul. Therefore, gk is a Jk-measurable, p-integrable

function on S.

Step 2. We claim that if a is a stopping time for

(Jk' k E N), then T is a stopping time for (Z n E N).

Let a be a stopping time for (Jk' k E N),

(a = k) E Jk. Therefore,

((Tk = n) ( (a = k) ZEn, k, n E N.

no
Thus (To=n) = ((Tk=n) (o=k)) c n n e N.
k=l

This proves that Ta is a stopping time for ( n, n E N).

Step 3. We claim that (gk' Jk' kE N) is an amart.

Let > 0 be given. There is some integer no such that

S f du f 'di < ,
Sf T 2'

for any two-bounded stopping times T,T' for (Z ,nEN), T,T'2n

Let T = limit Tk; then T is a stopping time for
k
(Zn, n N) and for each N1 N
f -,- f as k --t o.
k AN T AN as
k1 1
Since u has the *-property, from Theorem 3.3-4,

(f kAN J k' k N) is an amart.

We choose an integer K such that for every bounded

stopping times o, a' for (Jk' k N) with a, a' 2 K,










if fAN du f ANdui < E-

Let o, o' be bounded stopping times for (Jk, k E N)
and o, a' 2 K. Now

I fT VN d + I fTANld

= (f f dw+f fdi)+(+ f d + f dp)
(TN) C (TY
= {fT dii + f d.

Therefore,

1g du- fgdu = f d,- f odyI
f d+ f d- f d- f d
VNL T AN N f V f T G AQ
f IOVN1I T oVN I VAN1df- ANMd

< E.
This proves that (gk' Jk' k E N) is an amart.

Corollary 3.3-6. Assume that i has the *-property.
If (f, n, n e N) is an amart and a is a stopping time for
(z n e N), then (fnAc' Jn, n c N) is an amart, where
J = {AcE: Ao(nAo=k) E Ek, k E N}.

Proof. Let T = n A a, n N; then (T n E N) is
n n
a nondecreasing bounded stopping time for (Z n E N).
From Theorem 3.3-5 (fnAa' n n E N) is an amart.
nncy n
















BIBLIOGRAPHY


[1] D. G. Austin, G. A. Edgar, and A. Ionescu Tulcea,
"Pointwise Convergence in Terms of Expectations,"
Z. Wahrscheinlichkeitstheorie Gebiete 30, 17-26
(1974).

[2] R. G. Bartle, "A General Bilinear Vector Integral,"
Studia Math. 15, 337-352 (1956).

[3] P. Billingsley, Probability and Measure, John Wiley
& Sons, New York, 1979.

[4] R. V. Chacon, and L. Sucheston, "On Convergence of
Vector-Valued Asymptotic Martingales," Z. Wahrschein-
lichkeitstheorie Gebiete 33, 55-59 (1975).

[5] S. D. Chatterji, "Martingale Convergence and the
Radon-Nikodym Theorem," Math. Scand. 22, 21-41 (1968).

[6] N. Dunford and R. Schwartz, Linear Operations,
Part 1, Interscience, New York, 1958.

[7] G. A. Edgar, and L. Sucheston, "Amarts: A Class
of Asymptotic Martingales, A. Discrete Parameter,"
J. Multivariate Anal. Vol. 6, No. 2, 193-221 (1976).

[8] D. R. Lewis, "Integration With Respect to Vector
Measures," Pacific J. Math. 33, 151-165 (1970).

[9] P. A. Meyer, Probability and Potentials, Blaisdell,
Waltham, Mass. 1966.

[10] J. Neveu, Discrete-Parameter Martingales, North-
Holland, New York, 1975.

[11] R. W. Smith, "Convergence Theorems for Abstract
Asymptotic Martingales," Ph.D. Dissertation,
University of Florida, Gainesville, FL., 1979.
















BIOGRAPHICAL SKETCH


Lienzu L. Hsieh was born 1946 in Taipei, Taiwan,

Republic of China. She graduated from National Taiwan

Normal University in Taipei in 1970. She came to the

University of Florida as a graduate teaching assistant

in the fall of 1977 to study mathematics and received

her M.S. degree in December, 1978.

Mrs. Hsieh is married to John-tien Hsieh. They have

a daughter named Sarah Ellen who is nine years old.

Mr. Hsieh also received his Ph.D. degree in mathematics

from the University of Florida.












I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.





J. K. Brooks, Chairman
Professor of Mathematics


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.





A. K. Varma
Professor of Mathematics


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.





S. S. Chen
Associate Professor of Mathematics


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.


C. Burnap
Assistant Professor of Mathematics












I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.





R. B. Kershner
Associate Professor of English



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 Council, and was accepted
as partial fulfillment of the requirements for the degree of
Doctor of Philosophy.


December, 1981



Dean for Graduate Studies
and Research
































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