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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 REALVALUED ASYMPTOTIC MARTINGALES INTEGRABLE WITH RESPECT TO A REALVALUED 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 REALVALUED ASYMPTOTIC MARTINGALES INTEGRABLE WITH RESPECT TO A VECTORVALUED MEASURE .. 11 1. Basic Background and Uniform Integrability. 11 2. Decomposition of An Amart .. 27 3. Net Asymptotic Martingales .. 35 III VECTORVALUED ASYMPTOTIC MARTINGALES INTEGRABLE WITH RESPECT TO A VECTORVALUED 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 realvalued asymptotic martingales which are inte grable with respect to a vectorvalued measure whose values are in a locally convex topological vector space. Secondly, we treat Banachspace (or Bspace)valued asymptotic martin gales which are integrable with respect to a Bspacevalued measure. In Chapter I, we establish basic concepts concern ing realvalued asymptotic martingales integrable with respect to a realvalued measure. In Chapter II, the uniform integrability and the decomposition of realvalued amarts integrable with respect to a vectorvalued measure whose values are in a locally convex topological vector space are derived. Also convergence theorems of realvalued net amarts integrable with respect to a vectorvalued measure which takes values in a locally convex topological vector space are derived. In Chapter III, several theorems con cerning Bspacevalued martingales integrable with respect to a Bspacevalued measure are developed. We also extend some of these theorems and an optional sampling theorem to the amart case. CHAPTER I REALVALUED 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 subofields ( 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 scalarvalued amart converges almost everywhere if it is essentially bounded. Austin, Edgar, and Tulcea [1] proved that (*) "a realvalued amart converges almost everywhere if it is L1bounded." Chatterji [5] proved that if a Banach space E has the Radon Nikodym property, then every Evalued L1bounded martingale converges almost everywhere. Chacon and Sucheston [4] proved that if E is a Banach space with a separable dual and has the RadonNikodym property, then an Evalued 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 scalarvalued 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 realvalued 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 realvalued 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 scalarvalued 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 realvalued 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 Pintegrable functions f and g satisfy A fdy = /A gdp for every set A in the afield E, then f = g palmost everywhere. 2. Elementary Notations We shall begin by introducing basic concepts and notations for the realvalued amarts which are integrable with respect to a nonnegative measure X. A field of subsets of S, or aBoolean algebra of subsets of S, is a nonempty 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 ofield or Borelfield 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 ofield 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 zmeasurable 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 Emeasurable function defined on S, then the afield generated by f is the smallest ofield of S which contains all sets of the form (f < a) for any real value a. Similar, the subafield of Z generated by a family of Emeasurable functions (f aA), denoted by o{(f aEA)}, is the smallest afield 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 Xnull 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 Xalmost everywhere, or simply almost everywhere, if it is true except for those elements in some Xnull set. The phrase "almost everywhere" is usually abbreviated "a.e." A function f: S > R is said to be Xessentially bounded if 5 inf sup f (s) < N sESN where the infimum is taken over all Xnull sets N. A func tion f: S > R is a Anull function if (Ifl > a) is a Xnullset, 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 Anull function is called a Asimple function. A sequence (f ) of functions from S to R is said to converge in Xmeasure, 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 Ameasurable function on S if there is a sequence of Xsimple functions converging to f in measure. Two other convergence concepts for sequences which are often used in realvalued 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 Anull 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 L1integrable functions, we say that (f ) converges to f in mean if the sequence (f ) converges to f in L1norm. 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 Emeasurable 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 subafield 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 wellknown RadonNikodym 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 subafield 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 RadonNikodym theorem, there is a Jmeasurable, integrable function g such that E fdp = fEgdp for every E in J. A Jmeasurable 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 subafield 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 subafields 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 ffn 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 subafields 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 wellknown 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 subofields. 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 REALVALUED ASYMPTOTIC MARTINGALES INTEGRABLE WITH RESPECT TO A VECTORVALUED MEASURE 1. Basic Background and Uniform Integrability In this chapter we shall continue to develop the theory of realvalued 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 Psemivariation of the vector measure whose values are in a locally convex topo logical vector space V, where P is a seminorm on V. We shall start by introducing the theory of integration developed by Lewis, and the properties of realvalued amarts which are integrable with respect to a Vvalued 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 Vvalued 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 seminorm P defined on V. If x* e V* and P is a continuous seminorm defined on V we will write x* < P whenever Ix*(x) < P(x) for all x E X. If P is a seminorm on V, then the Psemivariation 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 2.11 [8, p. 158]. If P is a countably additive measure and P is a continuous seminorm 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 2.12 [8, p. 158]. If u is a measure, P a con tinuous seminorm and (E n N) a convergent sequence of sets in Z, then lp (limit E ) = limit Hl l(En) n n Corollary 2.13 [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 pnull set if it is a subset of a set E E Z such that lp (E) = 0 for every continuous seminorm P. Any statement concerning the elements of S is said to hold palmost everywhere, or simply almost every where if it is true except for those elements in a unull set. A psimple function f: S > R is a function which is ualmost 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 pmeasurable if there exists a sequence of Wsimple functions which converge to f palmost everywhere. A function f : S > R is pintegrable if (1) f is x*pintegrable 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 welldefined. It is obvious that the integral is linear n and for every simple function f(x) = c IE. f is pinte 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 uintegrable, then P[ Efdu] < l Ip (E). sup {f(s): s S} for each E E Z and continuous seminorm P. Theorem 2.14 [8, p. 160]. (1) If f is pintegrable, 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 pintegrable functions which converge pointwise to f on S and g be a pintegrable function such that Ifnl g for each n. Then f is uintegrable if V is sequentially complete. If f is pintegrable, then IEfdp = limit E f nd uniformly with respect to E E Z. n In Theorem 2.14 we can replace the hypothesis (fn) converges pointwise to f on S" with "(f ) converges to f ia.e. on S." The result is still true, because for each x* E V* there is a continuous seminorm 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*pintegrable for each x* e V*. The rest of the proof is the same as the proof in [8, p. 160]. Definition 2.15 [11, p. 60]. If (f n n e N) is a sequence of pmeasurable functions adapted to an increasing sequence of subafields (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 seminorm 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 seminorm P there is a o E T such that if T > u then sup { x*(ff dx x)I : x* < P} < e. Lemma 2.16 [11, p. 61]. Let (f ) be a sequence of Pintegrable functions adapted to an increasing sequence of subofields (Z ) of Z, and T (E) = IEf Td for each T E T. If sup {11 (S)II : T E T} < for every continuous seminorm 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 seminorm P. Proposition 2.17 [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 seminorm P. Proposition 2.18 [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 ofield consist ing of all intersections of elements of En with E. Definition 2.19 [11, p. 69]. A sequence (f ) of pintegrable functions is said to be Llbounded if sup {/ fn dv(x*Y) : x* < P; n E N} < for every continuous seminorm P. Theorem 2.110 [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 2.111 [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 2.112 [11, p. 72]. Let (f ) be a sequence of imeasurable functions adapted to (Zn, n e N) and suppose there is an integrable umeasurable 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 uintegrable function f. Lemma 2.113 [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 uintegrable function f ua.e. if sup { fn dv(x*u): n E N, x* P} < c for every continuous seminorm P. The next theorem constitutes an "optional sampling theorem" for amarts. Theorem 2.114 [11, p. 74]. Let (f En, n E N) be an amart and (Tn, n E N) be a nondecreasing 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 2.115 [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 2.116 [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 seminorm 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 2.117. Let (f ) be a sequence of pintegrable 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 seminorm P, where Inl1 (E) = sup {/ f dv(x*p): x* P}. np E n Definition 2.118. Let (f ) be a sequence of pintegrable 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 seminorm P. The next theorem shows that uniform integrability is equivalent to L1boundedness and uniform absolute continuity for a sequence of Uintegrable function. Theorem 2.119. Let (f ) be a sequence of pintegrable 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 seminorm 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 L1bounded, 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 seminorm. 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 2.11 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 2.120. Let (f ) be a sequence of pintegrable functions. Then (fn) is said to converge to a yintegrable function f in L1 if for every continuous seminorm P the following is true: limit fn fdv(x*p) = 0 uniformly in x* V*, x* < P. n f Note that from Proposition 2.11 and 2.14 it follows that if f is pintegrable and (E) = }Efdp, then II  (E) < m for each E in E and each continuous seminorm P. Lemma 2.121. Let (f ) be a sequence of pintegrable functions. If (f ) converges to a pintegrable function f in L then (f ) is uniformly integrable. Proof. Let P be an arbitrary but fixed continuous seminorm. 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 lki p(S) = sup {ifk dv(x*) : x* P} for each k = 1,2,...,nl1. This shows that (fn) is L1bounded. 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 wintegrable, there is a number do > 0 such that  p(E) < whenever E E Z and Ivlj (E) < 6S Now, IE fnldv(x*u) < Elfnf dv(x*Y) + jE fdv(x*P) E f nf 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 pintegrable, there exist 0 numbers 6',6 ,...,6 i such that for each m {1,2,... ,nol, 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 Mintegrable. This fact follows from the definition n of amart and Proposition 2.18. Definition 2.122. Let (f ) be a sequence of pmeasurable functions. Then we say (f ) converges to a pmeasurable func tion f in pmeasure, or simply in measure, if for every E > 0, limit iP l1 ({s E S: If (s)f(s) >C}) = 0 n for every continuous seminorm P. Definition 2.123. Let (f ) be a sequence of pmeasurable functions. Then we say (f ) converges to a pmeasurable func tion f Uuniformly if for every E > 0 and every continuous seminorm P there exists a set A in E such that Ifl (SA ) < 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 2.124. Let (f ) be a sequence of pmeasurable functions from S to R. If (f ) converges to a umeasurable function f pa.e., then (f ) converges puniformly to f. Proof. Let > 0 be given and let P be a continuous seminorm. Suppose that E is the pnull 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= ( (SE ) = limit (SEk ). k=l k It follows fromTheorem 2.12 that uIp (E) = limit I111 (SEk,m) k Since llp(E) = 0, there is an integer km such that II Ip(SEk < for each m. k ,m m Let A = E then S A = l (SE ) . m=l m=l m It follows from Proposition 2.11 that l (SA ) l(SEk ,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 puniformly. Lemma 2.125. Let (f ) be a sequence of umeasurable functions from S to R. If (f ) converges to a pmeasurable function f nuniformly, then (f ) converges to f in pmeasure. Proof. Let E > 0 be given and let P be a continuous 1 seminorm. Then there is an integer m such that E >  m Since (f ) converges to f puniformly, given any number 6 > 0, there is a set A in Z such that Ip,(SA0) < 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}SA for each k 2 k k m 6 m Therefore HpI {s: Ifk(s)f(s) I > E} l p (SA6)< 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 pmeasure. Theorem 2.126. Let (f ) be a sequence of nmeasurable functions. If (f ) converges pa.e. to a pmeasurable func tion f, then it converges to f in pmeasure. Proof. The result obviously follows from Theorem 2.124 and Lemma 2.125. Theorem 2.127. 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 pintegrable 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 pintegrable functions. It follows from Lemma 2.121 that (a) implies (b). We need to show that (b) implies (a). Since (f ) is uniformly integrable, by Theorem 2.119, we have that (f ) is L1bounded. Also (f n, n e N) is an amart. Therefore (f ) converges to a wintegrable function f yalmost everywhere from Theorem 2.113. Therefore (f ) converges to f in imeasure. Thus for any E > 0, limit 1 IL({SES: If (s)f(s) > ) = 0 n pn 3 H iI (S) for each continuous seminorm 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 iintegrable, 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 fnf dv(x*p) = _SA nfnf dv(x*p) + fA nfnfdv(x*0) n fn d 3p (3) I' (SA ) + 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 ff dv(x*p) = 0 uniformly in noo 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 realvalued amart (f ) for (Z ) integrable with respect to a Vvalued measure p, where V is a locally convex topological vector space, provided V has the RadonNikodym 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 2.21 [6, p. 165]. Let (S,Z ,) beameasure space on Z. Let Z1 be a subofield of Z, and p, be the restriction of w from E to 21. Then (a) convergence in pwmeasure implies convergence in pmeasure; (b) a p null function is a pnull function; (c) a p null set is a pnull set; (d) a p measurable function is imeasurable; (e) a plintegrable function f is pintegrable and E f d = IEf di for each E E 1. Since a function is pintegrable if and only if it is v(p)integrable, the result (e) of Theorem 2.21 still holds if A is a realvalued measure. Definition 2.22. A sequence of functions (f ), f : S > R for each n e N, adapted to an increasing sequence n of subofields (Z ) of Z is defined to be a martingale if the following hold: (1) (fn) is a sequence of pintegrable functions. (2) Sup { If dv(x*l): x* i P} < for each n E N and for every continuous seminorm 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 2.23. Let V be a locally convex topological vector space. Then V is said to have the RadonNikodym property if for every Vvalued measure space (S, Z, p) and every Vvalued measure v such that limit IvjI (E)=0 and IIvI (S) < for every continuous l I I (E)*0 P seminorm P, there is a pintegrable 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 RadonNikodym property. Let (f Z n e N) be a realvalued amart in the measure space (S, Z, p), where p is a Vvalued measure. Since (fn) is an amart, (f ) is a sequence of vintegrable 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 Ilp (E)O P p Theorems 2.11 and 2.14. 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 seminorm 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 RadonMikodym property, there is a k ,/Zmintegrable, Z mmeasurable 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 2.24. Assume that V is sequentially complete and has the RadonNikodym 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/ mintegrable function gm y/Za.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 dpiv ) : x* L P} = 0 TE T for every continuous seminorm 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*( IgTdvo) : x* P} TET ) m o m = limit sup { x*(Ifd dv ) : x* P} = 0 TET for every continuous seminorm 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 L1bounded, 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 mintegrable, 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*/pintegrable k from Theorem 2.21. Since (g k E N) converges to g mm p/Ema.e. it follows, that i Zml p({sES : gm(s) o g (s)} ) = 0 for every continuous seminorm 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 pa.e. Theorem 2.25. Let (f Zn, n N) be an L bounded amart and gm be a /Z mintegrable function such that (E(fk/Zm) : k e N) converges pa.e. to gm. If for each m N, gm is iintegrable and if there is a pintegrable, E1measurable 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' pa.e. and h = h' pa.e. for each integer n. Proof. (1) Since gm is pintegrable it follows that sup {/ gm dv(x*0) : x* < P} <  for every continuous seminorm 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 2.14. 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 ng 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 dv) : x* P =0 TcT f TET for each continuous seminorm P. Let vo = v gld then v E V. Therefore for each x* V*, x* < P we have o x*( h _dvo) = x*(f(fTgT)divo) = (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 jgldpvo) = x*( f dp v). Consequently, limit sup {x*( h dvo) : x* P} TT = limit sup {I x*( f dlv) : 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*/Za.e. because g ,g' are Z measurable. From Theorem 2.21 we have g = g' x*ia.e. for every x* C V*. Consequently, g = g' pa.e. for each n e N. This implies that h = h' ia.e. for each n EN. This completes the proof of the theorem. 3. Net Asymptotic Martingales Definition 2.31. Let A be a directed set and (f a a A) be a net of umeasurable functions from S to R adapted to an increasing net (Z a E A) of subafields 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 seminorm 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 seminorm P, there is a a E T which depends on e and P such that sup { x*( f dv) : x* < P} < E for each T ; T, T 2 0. Definition 2.32. 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 seminorm P. Lemma 2.33. 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 Umeasure, that is, limit (f f ) = 0 a ,a' A in umeasure, 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 uuniformly. Proof. Let P be an arbitrary continuous seminorm on V. Since limit (f f ,) = 0 in pmeasure, 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) kl k i=k p 2k1 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 ml 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 realvalue 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 SF. 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 iuniformly. 1 Lemma 2.34. Let (f a c A) be a net of pmeasurable functions. If (f a E A) is Cauchy in umeasure, then there is a subsequence (f i E N) of (f a A), and a pmeasur ai 1 able function f, such that (f i E N) converges to f ai 1 puniformly. Proof. From the result of the Lemma 2.33 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 iuniformly. We claim i that f is umeasurable. Since (f ) is a sequence of imeasur ai able functions for each i in N, there is a sequence of m m usimple functions (g m N) such that (g m N) converges to f pa.e. Hence (g m E N) converges to f uniformly 1 1 from the Theorem 2.124. Let P be a continuous seminorm. 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 SAk* mk Define hk = k for each k, then (hk, k E N) is a sequence of psimple 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 puniformly, (f ) con verges to f pa.e. Therefore, there is a set A such that pu p(A) = 0, and (f i) converges to f on SA. Thus Iulp (AUB) = 0. We claim that (hk, k e N) converges to f ua.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 pa.e. to f and f is a pmeasurable function. Definition 2.35. Let (f a A) be a net of pmeasur able functions. Then (f a e A) is said to converge to a umeasurable function f in pmeasure if for each E > 0, limit fIp ({s:SES, If (s)f(s) > E}) = 0, for every continuous seminorm P. for every continuous seminorm P. Lemma 2.36. Let (f a E A) be a net of pmeasurable functions defined on S to R. Then (f a A) is Cauchy in umeasure if and only if (f a a A) converges to a pmeasur able function f in imeasure. 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 umeasure, there exists a subsequence (fi i N) of (f a. A), and a pmeasurable function f such that (f i E N) converges to f in pmeasure. Let P be a continuous seminorm 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 pmeasure. The proof of the next convergence theorem is modeled after a similar theorem in Edgar and Sucheston [7, p. 206] for the realvalued case. Theorem 2.37. 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 seminorm P, then (f a, a E A) converges to a pmeasurable function f in pmeasure if V is sequentially complete. Proof. From Lemma 2.35, we need only to show that (f a E A) is Cauchy in pmeasure. Assume that (f Z a E A) is not Cauchy in pmeasure. Then there is some E > 0 and some continuous seminorm 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 dwv) : x* < P} < n holds for every continuous seminorm P. Therefore, there is some o E T such that 0 sup {}x*(jf dv) : x* < P} < holds for every continuous seminorm 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 anl and UI P ({s: Ifa (S)f l (s) > E }) ; o n ni if n is odd, then an is the one satisfying a 2 a n and sup { x* ( f dyv) : x* < P} < T n for every continuous seminorm P and each T 2 an, T s T. It is obvious that (f n N) is a sequence of pmeasurable 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 seminorm P, then sup { Ifa dv(x*u): x* P, n N} < ' n for every continuous seminorm 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 dpv) :x* < P} < < E for every continuous seminorm P. Therefore (f Z ,n N) n n is an L bounded amart. Therefore (f Z n N) con n n verges to a uintegrable function pa.e. and hence (f Z n E N) converges in pmeasure. But for every n n n E N we can always find some integer m, m > n such that 11  ({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 pmeasure. n n Therefore, (f Z a c A) is Cauchy in pmeasure and hence (f a a c A) converges to a imeasurable function f in umeasure from Lemma 2.35. CHAPTER III VECTORVALUED 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 Bspace, 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 semivariation 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 lpII (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 nonnegative 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 semivariation 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 Iill 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 pnull set if liE11 = 0. A proposition holds palmost everywhere, abbreviated as ua.e., if it holds outside of a inull set. A psimple function is a function f: S > X which takes only finite many values x ,... ,x each nonzero 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 3.11 [ 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 pmeasure 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 pmeasurable 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 palmost uniformly to a function f on S if for every e > 0 there is a subset A of S such that IA El < E and (f ) converges to f uniformly on SA . Lemma 3.12 [2 p. 340]. Let p be a finitely additive set function on Z to Y. (a) palmost uniform convergence implies convergence in umeasure to the same function. (b) ualmost uniform convergence implies ualmost everywhere convergence to the same function. We shall assume that p is a finitely additive set func tion from S to Y from Definition 3.13 through Theorem 3.110. Definition 3.13 [2 p. 341]. A function f, f: S>X, is said to be iintegrable over S if there is a sequence (fn) of usimple functions on S to X such that the following conditions are true. (1) The sequence (fn) converges in pmeasure 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 IEl< 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 3.13, then for each E in Z limit A (E) exists in the norm of Z. This limit is denoted nnco 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 3.15 [ 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 SE then Ffdu < E. Definition 3.16 [ 2, p.342]. A function f, f: S + X, is wessentially bounded on a subset A of S if inf sup f (s) < c, N seAN where the infimum is taken over all unull sets N. We write ess sup jf(s) for this number. sEA Theorem 3.17 [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 3.18. (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 pmeasure 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 threespace, 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 uintegrable; 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 pa.e. and in pmeasure 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 3.18 [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 3.19 [2 p.345]. Let (f ) be a sequence of pintegrable 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 3.110 [2 p. 345]. Let (fn) be a sequence of uintegrable functions which converge almost uniformly to f. Then f is integrable over any set E in Z with IEl < and limit fE fnd = jE fdy. n From now on we shall assume that U is a countably addi tive set function from ofield Z to Y. Under this assumption we can prove that if f: S + X is pintegrable, then the indefinite integral of f is a countably additive set func tion on Z to Z. Definition 3.111 [2 p. 346]. Let p be a countably additive set function on a ofield Z to Y, then p has the *property (with respect to X) if there is a nonnegative finitevalued 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 lip is countably subadditive on subsets of S. Theorem 3.112 [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 pmeasure to a function f, then some subsequence converges palmost uniformly to f. (b) If a sequence (f ) converges palmost everywhere to f, then it converges palmost uniformly to f. Theorem 3.113 [ 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 3.114 [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 pa.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 afield 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 subafields of Z with Z = a( U n ). n=l Definition 3.21. Let (f ) be a sequence of pmeasurable functions from S to X adapted to a sequence of subafields (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 pintegrable 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 3.22. A class W of subsets of S is said to be a Hsystem if it is closed under the formation of finite intersections: (I) A, B E W implies AAB E W. A class L is a Asystem 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 BA EL; (X ) A,A2,. .. L and An t A imply A E L We remark that a class that is both a Hsystem and a Xsystem is a ofield. Theorem 3.23. (Dynkin's 1iX Theorem) [ 3, p. 34]. If W is a Hsystem and L is a Xsystem, then W< L implies that o(W) C L. Lemma 3.24. Let p be a countably additive set function on Z to Y and assume that there is a nonnegative 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 Hsystem. n n n=l n=l We claim that L is a Asystem. 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 (BA) A (B A ) C (BAB ) U (AAA), v((BA)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 BA is in L. (3) Assume that A ,A2,..., are in L and An A. Let B = AA 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 = (AF ) 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 HX theorem we have 0( U 2 ) C_ Therefore L = o( LV Z ). n=l n=l Since Ifi 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 3.25. Let u be a countably additive set function on Z to Y and v be a nonnegative finitevalued measure on Z such that ii << v. Assume that (fn, n n e N) is a martingale. If there is a pintegrable 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 n1 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 ointegrable, limit gdp = 0, 1010 B there is a 6 > 0 such that ill < 6 implies IB gdu < 4. From lemma 3.24 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 mdpEf dp . A E Also, Ef dp f dp[ = i f d f dp E (EnA )U(EA ) (EnA )U (AEE) = fndp f f nd nd + f fndpI EA A E EA A E E E C SEgdl + I gdp . EA A E Since Ivi 1 is a monotone function on E, pi (EA ) p II (EAA) < 5 . Therefore if gdl~ < . EA 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 3.26. Let T be the collection of all bounded stopping times for (Z n N) and (fn n E N) be a sequence of vintegrable 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 (Tm+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 3.27. 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 3.28. Let u be a countably additive set function on Z to Y and v a nonnegative finitevalued measure on Z such that lu << v. If (f n' En n E N) is a martingale and there is a uintegrable 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 3.27 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 3.25 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 3.29. (VitaliHahnSaks) [6, p. 158]. Let (S, Z, A) be a measure space, and (XA) a sequence of Xcontinuous vector or scalarvalued 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 3.210. 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 pintegrable 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 3.25) (it is well defined from Theorem 3.25). 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 nonnegative, finitevalued 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. no n From the VitaliHahnSaks 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 3.211. Under the hypotheses of Theorem 3.210 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 3.210 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 realvalued random variable case which were proved by Austin, Edgar and Tulcea in [1, p. 18]. Lemma 3.212. Let (fn, n N) be a sequence of Xvalued, pmeasurable 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 Xvalued, pmeasurable 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 pmeasurable function, there is a sequence of psimple functions (h n N) such that h n h in pmeasure. 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 3.24 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 Mmeasurable. o o We claim that mo {s: gn (s)h(s) } C E ( j (AkEk 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 (AkEk)) = S; k=l k=l mo then s C U (AkEk) 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) 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(BBk)  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 nonnegative finitevalued 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 BB = k there is an integer k such that k v(BB ) < 6. Therefore pIl I (BBk )< o, o o p  (B) < l  (Bk ) + ii (BBk ) < 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 EMmeasurable, if n E N, M (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, ;Ip ({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 3.213. Assume that p has the *property. Let (f n N) be a sequence of Xvalued, umeasurable functions. Let Zn = o(f ,f2".. ,fn) for each n N and o( z E ) = n If h: S  X is a pmeasurable 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), pa.e. n n (2) If (f ) is a sequence of pintegrable functions, then h is uintegrable 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 3.212 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 n1 1 1 IIPII {s: If (s)h(s) 2 }  We claim that (f n N) converges to h in pmeasure. 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 3.112, there is a subsequence (Tkn, n a N) such that (f n E N) converges palmost uniformly to h, and therefore Tk kn converges palmost everywhere to h. (2) Since (f n E N) is a sequence of pintegrable functions, (f n E N) is a sequence of pintegrable func kn tions which converge palmost uniformly to h. Since p has the *property, lp II (S) < . Therefore, from Theorem 3.110, h is a pintegrable 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 Uinte grable functions f and g satisfy fdp = gdy for each B JB B in then f = g pa.e. This can be shown in the following. Example. Let X = Y = real Euclidean threespace, 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 3.214. 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 umeasurable 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 pintegrable 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 3.28, we have that limit fT di TET E exists for each E in Z. Also from Theorem 3.213, 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 3.31. Let (f ) be a sequence of pmeasurable functions from S to X adapted to a sequence of subafields (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 pintegrable 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 3.26, 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 3.32. 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 pintegrable function from S to X then sup { fd  i : E E L} < . SE From (gn, There Assume that f is a pintegrable function on S to X. Theorem 3.14, there is a sequence of psimple 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 3.17. 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 3.33. 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 uintegrable 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 pintegrable, 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 subafield Z such that jIIjI (AAE) < 6 0 from Lemma 3.24. 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 EA AE S2 gdu + 2 gdpl + 5 e. EA AE This proves that for each E in Z limit f dp exists. TT fE T Theorem 3.34. Assume that y has the *property. Let (f ) be a sequence of Pintegrable functions adapted to an increasing sequence (Zn, n N) of subafields of E. If (fn) converges Palmost everywhere to a Pmeasurable 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 ualmost everywhere to f, (fT n N) n converges ia.e. to f also. Since p has the *property, from Theorem 3.112, (f n e N) converges palmost n uniformly to f. From 3.110, 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 3.35. (The optional sampling theorem for amarts.) Let (fn' En, n E N) be an amart and (Tk, k E N) be a nondecreasing 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 Jkmeasurable, pintegrable 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 psimple functions (h m e N) such that (hm, m c N) converges to f in pmeasure, 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 Jkmeasurable 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 umeasure. 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 IIl 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 Ip 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 lll (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,< andif 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 Jkmeasurable, pintegrable 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 twobounded 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 3.34, (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 3.36. 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 3.35 (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, 1726 (1974). [2] R. G. Bartle, "A General Bilinear Vector Integral," Studia Math. 15, 337352 (1956). [3] P. Billingsley, Probability and Measure, John Wiley & Sons, New York, 1979. [4] R. V. Chacon, and L. Sucheston, "On Convergence of VectorValued Asymptotic Martingales," Z. Wahrschein lichkeitstheorie Gebiete 33, 5559 (1975). [5] S. D. Chatterji, "Martingale Convergence and the RadonNikodym Theorem," Math. Scand. 22, 2141 (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, 193221 (1976). [8] D. R. Lewis, "Integration With Respect to Vector Measures," Pacific J. Math. 33, 151165 (1970). [9] P. A. Meyer, Probability and Potentials, Blaisdell, Waltham, Mass. 1966. [10] J. Neveu, DiscreteParameter 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 Johntien 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 UNIVERSITY OF FLORIDA i13i 1 262Ii lI I08553 1111111111111111 3 1262 08553 6810 