MSU ‘ LIBRARIES —_—.. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES wil] be charged if book is returned after the date stamped below. MARTINGALES OF BANAOH-VALUED RANDOM VARIABLES By )A ((7)... x05 3. D1 CHATTERJI A THESIS Submitted to the School for Advanced Graduate Studies of Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Statistics 1960 ACKNOWLEDGMENTS It is with pleasure that I thank Dr. G. Kallianpur for having drawn my attention to the general area of martingales of Banach-valued r.v.'s. My sincere thanks are also due to Dr. C. Kraft for his kind encouragement to me throughout the period during which the research was conducted. The writer deeply appreciates the financial support of the National Science Foundation. MARTINGALES OF BANACH-VALUED RANDOM VARIABLES By S. D. Chatterji AN ABSTRACT Submitted to the School for Advanced Graduate Studies of Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Statistics Year 1960 M ri/ W/ 1 S. D. CHATTERJI ABSTRACT The main purpose of the thesis is to consider conditional expectations of r.v.'s which take values in a Banach space and to study the limit properties of certain sequences of such r.v.'s. These sequences are called martingale sequences, following the terminology of Doob. We first of all demonstrate that every Bochner- integrable r.v. has a conditional expectation relative to any Borel-field and establish some of the basic prOp- erties of conditional expectations. Then we go on to study the convergence in the mean and convergence almost- everywhere of martingale sequences. This we have done by studying operators on certain generalized Lebesgue- spaces, discussed in our Chapter 11. We have established the generalizations of most of the theorems of the classical theory of martingales and have shown by a counter-example in Chapter IV that some restrictions on the Banach space in which the r.v.'s take value, are necessary. In the last chapter, we have considered some applications of our theory. TABLE OF CONTENTS INTRODUCTION 0 O O O O O O I O O O O O O O O O O O l NOTATION, SOME DEFINITIONS AND KNOWN THEOREMS . . 6 CHAPTER 1 O O O O O O C C O O O O O O O O O O O O O 1 2 Conditional expectation of Bochner-integrable random variables . . . . . . . . . . . . . 12 CHAPTER 11 O O O O O O O O O O O O O O O C O O O 0 22+ Weak convergence in certain special Banach Spaces 0 O O O O O O O O O O O O O O O O O 24 CHAPTER 111 O O O O O O O O O O O O O O O O O O O O 40 Strong martingales of Banach-valued r.v.'s and their mean-convergence . . . . . . . . 4O CI-I‘APTER 1v 0 O O O O O O O O I O O O O O I O O O O 59 Almost everywhere convergence of Banach-valued strong martingales . . . . . . . . . . . . 59 CWTER V. O O O O O O O O O O O O O O O O O O O C 73 Some applications of the general theory. . . 73 BIBLIOGRMHY O O O O O O O O O O O O O O O O O O C 82 Introduction The notion of a measurable function defined on an arbitrary measurable space and taking values in another measurable space is a fairly well-known one in modern math- ematics. When the range space of the functions happens to have a topology also, various special concepts of measur- ability become important. Much research has been carried out, for instance, in the case when the functions take values in a linear topological space or more restrictedly in a Banach or a Hilbert space. A considerable body of the research is devoted to extending suitably the ordinary theory of Lebesgue integrals for real-valued functions. For functions taking values in a Banach space, there exist at least three different important concepts of measurability and integrability. This sort of extension of the theory of real-valued functions has been carried out in recent years in the study of random variables (r.v.) which after all are measurable functions on a finite measure space. Frechet (18)* considered r.v.'s taking values in a metric space and introduced notions of mean and variance for such r.v.'s. Does (19) considered r.v.'s taking values in topological ' Numbers in brackets refer to the bibliography at the end. 1 2 spaces with uniform structure and proved various generaliza- tions of the classical strong law of large numbers. Many other studies have been made with r.v.'s taking values in locally compact topological groups. But it seems that one can generalize the classical results of probability theory most satisfactorily only when the range space is at least a linear topological space for then much of the usual integration theory remains valid. In this direction, pioneering work was done by Mourier (10) who considered the range space to be a Banach space and not only proved some strong laws but also studies characteristic functionals of r.v.'s taking values in Banach spaces. Since then quite a few papers have been published concerning general strong laws of Banach-valued r.v.'s, e.g. Beck & Schwartz (1}), Beck (20).’ However, to the best of the author's knowledge, no more than one attempt has been made to define an extension of a basic concept of probability theory, namely, the con- cept of conditional eXpectation of a r.v. taking values in a Banach space. Beck & Schwartz (13) do define a notion of conditional expectation that we have used here, but they did not make any attempt to prove its existence. Dubins (21) defined a conditional expectation of a more general nature than ours but the difficulty with his defini- tion is that it does not yield an exact analogue of the standard theory. There is a basic difficulty in the process of defining conditional expectations for r.v.'s taking values in spaces like Banach spaces. That difficulty is the non— existence of a general Radon-Nikodym theorem for set func- tions taking values in non-compact spaces. The definition that we have used circumvents this by considering Bochner- integrals, for which although a general Radon-Nikodym type theorem is not valid, much can be done owing to the simple structure of integrable functions. Our main purpose here is to study this particular notion of conditional expectations for Banach-valued r.v.'s and then use this definition for considering generalizations of martingale theory for Banach-valued r.v.'s. One of the most important considerations in the study of martingale theory of scalar-valued r.v.'s is that of convergence of the martingales. We have studied this for the case of Banach-valued martingales specially from the point of view of treating conditional expectations as Operators on suit- able Banach spaces. For instance, our mean convergence theorems in Chapter 111 are reminiscent of the work of Lorch (22) concerning monotone sequences of projections on a reflexive Banach epace. Our results on the mean con- vergence of martingales, specially, have been obtained by simple linear space methods which are different from Doob's (1) approach. For proving almost-everywhere convergence we have used a generalization of a theorem of Banach and thus shown how many of the properties of martingales are simply the properties of a type of sequence of operators on a Banach space. In Chapter 1 we define our conditional expectation and prove its existence and general properties. In Chapter 11 we prove for future work weak compact- ness prOperties of certain Lebesgue type Banach spaces, some of which at least (e.g. Th. 2.3.1 and Th. 2.4.2) are not to be found in current literature. In Chapter 111 we consider the mean convergence of Banach-valued martingales. We prove the most general mean convergence theorem here under the assumption that the Banach space is reflexive. As shown by a counter-example in Chapter lV, it is clear that some such restriction on the Banach space is necessary. In Chapter IV we consider the almost everywhere con- vergence of Banach-valued martingales. We prove three different types of theorems, some using a theorem of Banach, one using Doob's idea of optional stopping and one using results from standard martingale theory. In Chapter V we consider two different applications of the theory, one to the study of the strong law of large numbers for Banach-valued independent identically distribut- ed r.v.'s and the second to the study of derivatives of Banach-valued measures with respect to nets. An attempt has been made to construct as far as was possible, a theory based only on linear methods. It is hOped that in the future more powerful linear space methods Ul will make the phenomenon of convergence of martingales of Banach-valued r.v.'s quite transparent to our com- prehension. Notation, some definitions and known theorems Let Q be an abstract set of elements or points w . Sub-sets of Q will be denoted by upper case Latin letters like A, B, F etc. Given two subsets A and B we shall mean by A c; B : A contained in B B CA : B contained in A A U B : union of A and B A n B = intersection of A and B A0 = the complement of A A - B = A A BC AAB : (A-B)U(B-A) 95 : the empty set If ACE and BC, A then we shall write A = B . The symbol "5" shall denote the relationship of an element belonging to a class. We shall occasionally use the symbqls '"3" and "a" as short-hand for the phrases "such that" and "there exist(s)" respectively. By a fieldif sets inflwe shall mean a class of sub- sets such that 1) é andfl are in g’ 11) If A 53!" , then A0665 111) If Aie 9: , i = 1, 2, .. n where n is a finite postive integer then One? 1:1 7 By a Borel-field of sets in n we shall mean a class of subsets 3' of .0- such that 1) (P andfl are in 1; 11) If A 6‘3; then A069; 111) If A16 63; i = 1, 2, ...( a denumerable sequence of sets in ‘3"— ) then Lg A1 6.3; A "probability space" will be a triple (Q, g , P) where I)- is any abstract set, S; , a Borel-field of sets in (1., and P is a non-negative function defined on SF such that 1) P(A) 7,0 for all A69: 11) If A163 1:1, 2, then P( L! A1) = 10%.1P(Ai) 111) 14.0.) = 1 . By a "Banach space"xover the complex numbers (for brevity, Banach space) we shall mean a set of elements which is such that 1) It forms a vector space on the field of complex numbers 11) There is a function « x|| defined on 36 , called norm such that IVA-X“ = I‘M-"x. “any complex number ux+yug 11 xii + n 111 “x N = 0 if and only if x = O (the zero element of the vector space) 111) for any sequence x1, 1 = 1, 2, ... of elements of x for which 11m -X“:O m,n—-)oo“xm n there exists an element xe 36 such that lim x-x\\=O n—?00“n A complex-valued function x* defined on f such that x* (x + y) : 1* (x) + x* (Y) x* (2.x) = ).x* (x) ’A any complex number (1* (x)|é A. ll xll for some 'A >,O and all x e x will be called a bounded linear functional on x . Occasionally we shall use the notation (x, x*) for x* (x) . With I1X*|\ = supg |x* (X)|; llxllsl} the set of all bounded linear functionals on I forms a Banach space x" called the "dual" or "conjugate" off . We shall denote by $9” the dual of 1* i.e. x4“ : .x. *- (se) . «H- * If we consider the function x (x ) on f de— fined by x**(x*) : x*(x) xex , x fixed. then x** is a bounded linear functional on 1* with u x**ll = 11x11 If all the bounded linear functionals on 35* are of this type then we shall write % = %” and call .36 a "reflexive" Banach space. A sequence of elements xn e x is said to be "weakly convergent" to x e x if lim x*(xn) = x*(x) n—aoo for all x * E f. A set of elements S C} will be said to be "weakly compact" if for any sequence of elements xneS there is a subsequence of elements x which converges weakly to n.) some element x which may or may not belong to S. (Actually, in standard theory, this is called conditionally, sequentially, weakly compact. But because we shall not have occasion to use any other kind of compactness, there- fore we prefer this briefer expression. However, the works of Eberlein and Phillips (see Hille & Phillips (3) pp.37) show that in many cases our definition of weak compactness is the same as the notion of compactness under the weak topology of I which we do not discuss here.) The following theorem of Pettis shall be used often: (For proof, see Dunford & Schwartz (2), pp. 68-69). A set S in a reflexive Banach space is weakly com- pact if and only if it is bounded i.e. { u x“: x e S} is a bounded set on the real line. A reflexive space is weakly complete i.e. whenever a sequence xn of elements is such that lim x* (xn) 11-—)di )0 exists for every x* 3 1* there exists an element x such that xn's converge weakly to x . A bounded linear operator T from a Banach space % to a Banach space ’3 is a function on E taking values in ’U such that 1) T(x + y) = T(X) + T(y) 11) T(’)\.x) = ’A. T(x) 111) “Txflé A.\|x|| A),o , and x635 . We define I) T" : sup{l|Txll : || xné I} The following is sometimes called the Banach- Steinhaus theorem: "Let ‘GE, Qj be Banach-spaces and {TnN} be a sequence of bounded linear operators on SE to Qj . Then the limit Tx : lim Tnx n -—) oo exists for every x E. 96 if and only if i) the limit Tx exists for every x in a everywhere dense sub-set of :2: ii) sup N Tn xu<+ao for each xe x n When the limit Tx exists for each x e if , the Operator T is linear and bounded and “T “ 4 L192 \\ Tnngsuputnm + .o". n-9ao wt (For proof: See (2) Dunford and Schwartz, pp. 60-61.) - If .XKO) is a function on a probability space (.0. , r8 , P) taking values in a Banach space x then Xko) is said to be strongly measurable with respect to 11 7811" lim xh(u» : Xk») a.e. (everywhere except rr—900 on a set of points of P - measure 0) where xn(w) :-1 : 31(n) ’K (?)(0) E sin) e, % 12”,” 5’18 , Ein) n Egn) =95, LijE(1;1):fl «Fa.» : 1 if £06 F for any set F in .fl. . :0 if (.9ch Define Xn(w) to be integrable if .i|1a(n)ll P(E(il)) L=l (+ co and write an(w)dp :1 53(3)“) gin) We say that XKB) is Bonchner-integrable if there exist a sequence of integrable functions Xh(a» as above such that 11111 X (00) : X(60) a.e. n—awo n and m, n—§mo Then it follows that lim N S Xn(co)dp - 5Xm(w)dp” : 0 lim 5“ Xn(co) - Xm(w)“ dp = O . nn2na7oo Hence lim X (0) dp rr¢7q> n I I (1’ 1A exists and this we define to be ‘5. Xce)dP, called the Bochner-integral of X((.)) , occasionally denoted also by us as E(X«0)) or, E(X) . It can be shown that X10» is Bochner-integrable if and only if i) X(co) is "almost everywhere separable- valued" 1.6.3 NEE , P(N) = 0 such that the set sci defined by s ={ X(o.1) : me No} has a denumerable dense sub-set. 11) Emma)" ) = (u x1| dP< me (If (1) is satisfied then H X(w)u is automatically a non-negative function measurable with respect to GI; .) For a discussion of Bochner-integrals see (3) Hille & Phillips, pp. 71-89. The notion of "uniform integrability" of a family of complex-valued measurable, integrable functions Xt((d) t e T , on ( .{l , ’63 , P) shall be defined as follows: For any e) O , there is a 37,0 such that 1(”xt( )"1 dP< e for all t e T .,A Whenever P(A) < q . In the text a theorem, lemma or equation numbered 'u.v.w. where u.v.w. are positive integers will be the w th one in v th section of u th chapter. Chapter 1 Conditional expectation of Bochner-integrable random variables § . Let ( .O. , {B , P) be a probability Space i.e.«Q— be an abstract set of elements so , (B , a Borel-field of sub-sets of £7« , called measurable sub- sets and P(-) a countably additive, non-negative set- function defined on (E3 and such that P(-(1') = 1 . Let Cf: be a Banach-space. We shall denote by H x" the norm of an element x e: :35 and by f1?* the dual of 7“ . A function X(w) defined on Q- and taking values in :36 which is strongly measurable with respect to the Borel-field ’63 will be called a GE; -valued strong random variable or when there is no SCOpe for confusion simply a random variable (r.v.). Let E; be a Borel-field contained in {ES i.e. ‘3' C If?) and let X(w) be a r.v. which is Bochner- integrable. Following Doob (1, pp. 17) we shall define the strong conditional expectation of X0») relative to or given (3' , in symbols E(X ‘g) as follows: 22;: 1.1.1. E(X '31) is a x -valued Bochner-integrable r.v. strongly measurable with respect to the sub-Borel- 12 13 field. S}; (for brevity a E; -meas. r.v.) such that for every A 6 31’ it satisfies the equation Smxrf) dP: f XdP . . . . (1.1.1) A A where the integrals are taken in the sense of Bochner. We shall now prove the existence and uniqueness of E(X I?) for every Bochner-integrable r.v. X(w) and Borel-field '39 C "fl . The standard proof for scalar-valued r.v.'s Xko) cannot be extended to cover the situation here since the analogues of the Radon-Nikodym theorems for set-functions taking values in a Banach-space are not in general valid. For counter-examples see Bochner, (7) Clarkson (5). Theorem 1.1.1 E(X \?5) exists and is unique except for sets of measure 0 for every Bochner-integrable r.v. x(co) and any Borel-field (E): C (B . (Notice that no assumptions on the Banach-space LEE: are made.) Proof: We shall use the fact that X(w) being Bochner- integrable is almost everywhere (a.e.) separable-valued and E(I1X(GD)||)< +00 . (3, Hille pp. 80) Because X(co) is almost everywhere separable-valued we might and shall without loss of generality consider f to be separable; for otherwise, we can carry out the proof by restricting our attention to the separable sub-space in which the values of X(00) lie with probability one. 2% being separable 14 there exists a denumerable determining set (3, Hille pp.34) i.e. linear functiomfls x; 6. 3E* i : 1, 2, ... such that for any x e f we have "x”: 83pm; m1 Of course it follows that llxillls 1 From equation (1.1.1) it follows because of an elementary property of Bochner-integrals that for every x’e 2* x*{ 10431)} = E(x* (X(w))\3') a.e. We shall first show that if E(X ‘2‘) exists then it must be unique except for sets of measure 0 i.e. if Y1(w) , Y2(€J) , are r.v.'s which satisfy def. (1.1.1) then YIQa):: YQWD) a.e. This follows because SYMQ) dP : SY2(w) dP A A for all A E §F and hence 41' S * x Y dP) = x (SY dP) . ( A, . .2 * _ i.e. SAXi (Y1 - Y2) dP - 0 Now Y1, Y2 being E; -meas. so is Y1 - Y2 and hence xi* (Y1 - Y2) is E? -meas. It follows from a standard theorem in measure theory that * _ _ X1 (Y1 Y2) - O a.e. 15 Hence *- X1 (Y1 - Y2) = o for all 1 = 1, 2, ... except for 606 N . P(N) : 0 so that 11‘ we N .. 41- 11111.1) - 12m)" _ exile l x1 (Y1(a>) - 3,20,), = O i.e. Y1fln) : Y2(G0 . This proves the uniqueness of E(X lEfi) . We shall now prove the existence of E(X '25) . To do this we consider two different cases: (1) Xku) is countably-valued i.e. X(co) = oi an 9‘11; (£0) with E(ll M») II ) = n:1 n 2-1 11 an“ P(En) < + 00 . n: where anE x, EnerB 9 “ (w){:1 wEEn n:1,2’ En o ‘0ng and En's are disjoint. N Consider YN(B) : “‘ anfnéd) where n: "1 HEAT) = E< '79: (MW) i.e. a conditional probability of En relative to i; , rut») and let N XNM = 2. an ’XEnoo) n:1 Then x1 (YN(w)) = Zx * (an) . fn(co) N 1:131 E(xi'x' (an) (XEn(w)\r§') = E(§.X1* (8'11)er (NIB?) n:1 n Hence for N‘7vM , using standard properties of con- ditional eXpectations of scalar-valued random variables we have 4;. N 31' [x1 (Y1: - 1M” elmng x1 ((1.0%, (”‘65) _ M * 7( n M: we.) Enter): )1 n:1 n:M+ 1 N 5 EH 5‘ X: (“.méwnl‘m n N e E(ni1uawn/XE11‘13‘) '-" E( II XN " XM" lg ) Hence 1) YN(‘°) " YM(“’)" = mi" 1 XIWN " YM)‘ L x _ . . -EH1N xMul'1) (112) Now lim X - = O a.e. and M, N——)oo H N KM 11 \1XN "KM“ €: 2 H.X|1 a.e. From whence lim E( || — X 111; ) : O a.e. (1 Doob CE M, N.._)oo KN M pp.23) 5 00 Hence the series Y((0) : n; a.“ fn(w) converges strongly a.e. 17 It is also clear that ‘[ Y.dP : 5 .X dP for all A ..u (1.1.3) and hence YN : BUST I 6}) according to Def. 1.1.1 Now XN(co) —9 X(co) a.e. I1XN(0°)“ é X(w) which is integrable -«-(1,1,4) YN(GO) —-) E(w) a.e. L (‘1‘ 111N1~ E<1XN1|S >4. E<11x11 m 1 which is integrable. Hence applying the dominated convergence theorem for Bochner-integrals (3, Hille pp. 83) we have by passing to limits as N -—9cn on both sides of (1.1.3) deP: )xar A A Also Y0») being the a.e. limit of EMU») which are E; -measurable r.v.'s is 1; -meas. Hence we have proved that no 11(10): Z1 an NBA?) = E(X "31) a.e. n: It is also clear from (1.1.4) that "1(a)" éEOIXlll‘iFI) a.e. (1.1.5) (ii) Let X03) be an arbitrary Bochner-integrable r.v. Let an be a denumerable dense set in :kg . Then for any k = 1, 2, 3, .... 18 co (1. = U)“ u x(w) - an 11 451/1} 11:1 w 219.. Sn, k Define Xkflb) k : 1, 2, as follows 60) : 8. Q S = a we 3 (1 SC 2 2, k 1, k _ a s so nsc etc ‘3 “€3.k"1,k 2,1: ' Obviously for all GO 6 Q (I Xk(a>) - X(0) H {— 1/k - - — (1.1.6) ll ywmé ll xl| + 1/k and hence Xk(w) is Bochner-integrable. Let 1km) = E(ka’o which exists according to the proof of case (1) above. For n7 m , and any x: of the determining set we have 1‘ (1n- Ym)‘ = 1 x: (sung) .. E(xmfi.) )1 =1E(X* (Xn) l?) - E(X: (Xm)|'5")la.e. |x |E( x: (Xn - Xm)| '3') )1 a.e. é E(|x: (Xn - Xm)ll 3?) a.e. é E(IIXn - Xmll '3!) a.e. Hence || 1n .. ym“ = 311111le (1n - ym)| é 13(11):n - 3811",?) a.e. éE(lan—xfll'}i)+s ("Km-X11159) a.e. 19 4. 1/n + 1/m a.e. because of (1.1.6) Hence lim " Y, - Y n = O a.e. so that m, n —+)d> ‘1 m lim Yn (co) =x(w) exists a.e. ...(1.1.7) n ——)d) From (1.1.5). (1.1.6) and (1.1.7) we have now a.e. 11 :X0 anXnW) () Imam)" 4.- ~HXW)" +1 lim Yn(t.>) = 1(a) n -—9a) 11 1.11am 4 E(|1 anl ('3' )9 E1 "xv-1)" +I I?) = E(l|x(o)ll |%) +1 Also because Yn : E(Xn'fi) we have for all A5? I gar: {YD dP ......(1.1.9) A ----(1.1.8) Because of (1.1.8) we can pass to the limit as n -——9oo on both sides of (1.1.9) invoking the bounded convergence theorem of Bochner integrals (3, Hille pp. 83) thus obtain- ing S‘ X dP : J; Y dP A A . This then proves that 1(60) : E(X (‘39) .and completes the proof of the theorem. s 2. Properties of strong conditional expectations: Almost all the prOperties of conditional expecta- tions of scalar-valued r.v.'s (1, See Doob pp. 20-26) can be established for the general Banach-valued case and 20 their proofs can either be obtained by mimicking the usual proof or can be derived from the scalar—valued case. In the following we shall mention a few of the standard properties. Theorem 1.2.1 If g; is a Borel-field such that A E 6 implies that there is B E (.3; such that P(Azl B) : 0 then E(x(c.>) (‘33) = E(X(u) [Q ) a.e. Proof: Let {xi} be a denumerable determining set for the X(w) values in f . Because of the validity of the theorem for scalar r.v.'s we have x; C E(X 19)) E(x:(x)| 3:) a.e. E(x:(X)‘ g) a.e. xi*(E(X‘$) - E(X ‘65)): O a.e. for Hence each 4.. so that ((111me - E(Xlé)“ : 8111p \x:(E(x|33) — E(x|§))\= o a.e. This proves that E(X)?!) : E(X'g) a.e. Theorem 1.2.2 Suppose 6." C g2 are Borel-fields and that some version (and therefore every) of E(X(¢D) ' g 2)‘ is measurable Q1 . Then E(X(w)|@‘) = E(x(w)|@2) a.e. 21 Proof: Let {x1*} be as in the previous theorem. Then x1*( E)) is Bochner-integrable (3, Hille pp. 84) and by a standard theorem for Bochner integrals (3, Hille pp. 83) I Tx(u>) d P = T(5A mo) dP) A6,? A Sfence A Tx(w) dP = T(_); X(w) dP) = T(_{E(X ‘3») <1?) = JAT(E(x|$)) dP Also T E(X I?) is 3-measurable so that according to Dan 1.1.1 we have T(E(Xl3"1)) = E(T(X(w))| ‘5‘: ) a.e. Theorem 1.2.5. If g C c; are sub-Borel-fields of m then Human?) lg ) = E 1g ) Proof: Follows directly from definition 1.1.1. Chapter 11 Weak convergence in certain special Banach spaces. §1. For our later investigations we shall need to know some properties of weak convergence in certain Lebesgue-type Banach spaces, first introduced and systematically studied by Bochner and Taylor in 1938 in (8). We define these Banach-spaces as follows: Definition 2.1.1 We define Ltd), ’8 , p, (f) 1 5 P<+ao as the set of all equivalence classes of strontly measurable r.v.'s X(Cd) defined on the probability space (.0, ’B , P), taking values in the Banach-sapce ‘35 and such that the "norm" [mm 31» = (£11m) 111’ dP)1/t < ..o ......(2.1.1) The equivalence class gxflo) } is set of all r.v.'s E(co) such that E(G) = X(w) a.e. Definition 2.1.2 Lao(Q. , ‘B, P, If) is the set of all equivalence classes of strongly measurable r.v.'s X¢D) defined on the probability space (a, '8 , P) and taking values in the Banach space if and such that the norm [male : escs);gs-up.\\X(w) “ ( +ao (2.1.2) 24 25 It can be shown that with the norms defined by (2.1.1) and (2.1.2) the spaces Lyn, ’B , RI) 1g 1’ 5 do are Banach spaces (2, Dunford and Schwarz pp. 146). When there is no danger of any confusion we shall abbreviate Lyn, "(B , P, x ) as LP( g). If ‘36 is the Banach-space of ordinary complex numbers we shall simply write LP . We shall invariably write X69) for the equivalence class ‘2 X(o)} . Our main objective is to study the weakly compact subsets of Lyx), 1 l; P(oo under the further assumption that x is reflexive. This we do in three steps. In § 2 we settle the problem completely for pj71 with the help of known results and give a representation for linear functionals on L1(3E) . In § 3 we study weak convergence of sequences of L1(3E) to an element in L1(I) and in g 4 we give one necessary condition and one sufficient condition for a set in L1(SE) to be weakur compact. §Eh The linear functionals of the Banach space LP(JE)' 1 4 P(oo have been studied by Bochner and Taylor 1938 (8), Day 1941 (14), Phillips 1943 (9). Dieudonné 1951 (15), Mourier 1952 (10), Fortet and Mourier 1951 (11). Bochner and Taylor gave for an arbitrary 5E: a general representation for the linear functionals in terms of certain Stieltjes integrals with vector-valued measures. 26 Under some conditions on the Banach-space E , they and others have also given simpler integral representations. We shall mention one such result due to Phillips for the case 1 < p (00 . Theorem 2.2.1 Let x be reflexive and 1

), X*(m)7 dP (2.2.1) where X*(€O)€Lq(Q,B , P, (£18,113 + = 1 1 5 ~21- and < x, x > denotes the value of the linear functional x* at x. 1/ \| F I) = (5)1x*(¢°)llq dP) q so that (Lg-0., r8 , P, 1))” is isometrically iso- Also morphic to the Banach space Lq(Q , '8 , P, x*) . Corollary 2.2.1 If x is reflexive then the space LP(Q , (B , P, f )’ 14 p (00 is weakly complete and a subset of it is weakly compact if and only if it is bounded in Lp(1) - norm. 2390:; It follows from theorem 2.2.1 that if if is reflexive then Lp(% ) is also reflexive (Notice that the converse is true also) and the corollary follows from standard theorems about reflexive spaces (see 2, Dunford & Schwarz pp. 68-69). 27 Mourier, 1952 (10), proved essentially the same result and Bochner & Taylor (8) proved the above under a condition on SE which is more general than reflexivity. Fortet & Mourier 1951 (11) proved a similar result for p 2 1 under the assumption that Iis separable. We shall need an extension of theorem 2.2.1 to the case p = 1 for our future work and shall in the following give a simple proof for it using a theorem of Phillips 1943 (9). Theorem 2.2.2 If F(-) is a bounded linear functional on L1( Q ,6 , P, 36 ) and x is reflexive then F(X) =f). Y*(w)) dP (2.2.2) where 1*(~)€ L‘(Q , ’8 , P, f‘). m For fixed E e'B , 81.7%») e L1(x) for all a 6x . Consider P(a.XE(u)). Because F is a bounded linear functional on L1( 2) we have I F(a°))l e 11 F II-Ea’XEwD. =11 111.551 an d? =||F||.I|all. P(E) o o o 0.0 (20203) Hence F(a. (XEGAH, for a fixed E 6% , is a bounded linear functional on x and let us write '3? * {- FCa. 985(0)) : xE(a) where XE e x / Also from (2.2.3) we have that 28 fi- 11- "xE u = 811113119) x E(a)‘ 5111.111. P(E) ..(2.2.4) * * * It is also clear that x EUF : x E + XF , EnF=4> .. E. F. e ’(B . In other words x; is an additive 3E* -valued set function on ’8 having the property (2.2.4). Accord- ing to a theorem of Phillips (9) there is'a function Y*(00)€ L‘o( x41) such that x; = SE 1*(0) dP. so that F(’XE(o).a) : $15 a, y*(w)> dP Hence for all simple functions n a1€EBE Xm) = 1; 611in (w) Eié'B we have * P(X(m)) = 501m), 1 (1.1)) dP . . .(2.2.5) If X(6)) € Lflz) is an arbitrary function then we can construct a sequence of simple functions Xn(Q)e L1(%-) such that lim Sumac) — 1111(0)“ dP = o n —-§¢’ (2, Dunford & Schwartz pp. 125) Hence from (2.2.5) we have p(x(w)) -_- iii-:00 P(Xn(u)> n n Ergo 34’9“”): r*(a1)> ‘1? 50m») . Y*(w)> dP 29 8.8 Karim), 1*(ao)> dP -f< X(0). Y*(°)> dP) ‘ 5. (11 x1») - ngll. 11%)" d? g [1*(u)]m15” x(c.>) - we)" dP —-) o as n —-31 00. This concludes the proof of the theorem. §13. Conditions under which a sequence of r.v.'s Kid“) 6 LNX) converges weakly to a r.v. X(C.))€ L1(1) were given by Bochner & Taylor (8) when E is of a special type. Our theorem 2.3.1 is of a different nature although the conditions involved are similar. Theorem 2.3.1 If Xn(CJ)€ L1(.Q, (B, P, X) , :BE reflexive, is weakly convergent and if “ Xn“ is uniformly integrable i.e. given 9) O , 3 5' 7 0 such that )llxn" dP) converges weakly to X(d>). Proof: We use a Radon-Nikodym type theorem due to Dunford and Pettis, 1940 (6) which can be stated as follows in our case: Let :36 be the adjoint to a separable Banach space {H and let X(E) be defined from f6 to E . Suppose that (i) for each y 6 3 the set-function XE(y) is completely additive 30 (ii) XE(y) : 0 when P(E) : O for all y (iii) the numerical function 0" = sup 1 xfi(y)| = | x II E I II II | 1‘ l E has finite total variation on any set E166 then there exists X(CO) e L1(§) such that XE = S mm) d? E Because {X113 is weakly convergent, it follows from a general theorem that E X ].(c . nl'; ' According to the representation theorem of linear functionals of L (:1) (Th. 2.2.2) we have that nLlim $5< KM“), Y*(Q)) dP exists for all Hi! *(0) 6 L‘o( IX, *) Take Y*(OO) : ,XE(0).a*, a*e 1* , EE’B- Then we have that as 11m a ( S Xn dP) exists for all n —900 E *E 1* and hence because 2% is reflexive the limit is a*CQ(E)) for some 2(E)ex Now i being reflexive we have 1: (gf‘j‘ and since we are concerned only with { Xn(b))} we might as well consider x to be separable. Then 1* 31 would also be separable. We shall now show that under the hypotheses of the theorem “(33) satisfies the conditions of Dunford & Pettis theorem. » a i) Let E166 . UEizE. Then *- °° * - d x(Q(L:E1)) —nl:u_1;°ox (SGEXn P) 1:1 1 = lim lim {(5. X dP 119‘ m —-)ao 13% E1 n We now show that m 300 EEK-(11.31 E1 Kn M)XTUE1)%dP) uniformly in n because *(I x. dP)\ e Mum“ an cu» m .9” 1:1 Verification of (ii) is quite trivial. iii) We shall show that for any finite number Of sets E1 1:1, ooooooooN E1 disjoint N :6} < 0 i=1 1 * N N ‘X(Exn dP). : lim‘ n [n lenll dPélim[XnJ1) e L1(X) such that ’)\(E) = In») dP E and hence j dP ———) fl é. g1: [x1 u . || jxn dP" E E1 1 L M+p A. — 3;: uxn u a? E1 .4 A- uxnndP M+p U E M i for all n and p if M)Me because “XII are n uniformly integrable. 34 Now an arbitrary 2*(0)E Lab (is? ) being a * uniform limit of countably-valued functions in H¢>(3E ) we have in general SdP —-) f°° En 35 X ‘5 K, x*(E.3€ 23923: We shall use the following generalization for vector-valued measures of a theorem of Nikodym (2, Dunford & Schwartz, Th. IV. 6. pp. 321) namely, "Let {ft-1} be a sequence of vector-valued measures defined on the Borel-field ’53 . If [‘(E) = n13}; h“) "I! exists for each E E (B , then I“ is a vector"o measure on ’B and the countable additivity of rm is uniform in n " . If K is weakly sequentially compact, then from a general theorem it follows that K is bounded. If (ii) 00 is not satisfied then 3 G > O, Eng'fi, J, , C‘En =¢ x* e 'I“ and xneK such that zvcgnxnm e We may assume {Kn} to be weakly convergent since K is weakly sequentially compact. Hence S Xn dP converges weakly to a limit for E each EEIB as * ’X X X. E dP is a linear functional on * * L1(I), for any X E i But then (:> is a contradiction to Nikodym's theorem. This proves the theorem. Theorem 2.4.2 Let KCL1(-(1, (B, P, I), 36 % reflexive be such that 1) [x140 for all xex, 0 independent of X 11) {“xw)“ : x e K} is an uniformly integrable family then K is weakly sequentially compact. Proof: We shall need a lemma due to (2) Dunford & Schartz, pp. 202. Lemma: Let’B be a Borel-field of sets and m, a field contained in (B which generates (a . Let {kn} be a sequence of countably additive set functions on ’8 with values in x . Suppose that the countable additivity of f- 1s uniform in n and that lim film) 11 n a“ exi t for E . Then lim (E) exists for ss 6’81 n-ern E e (B . Corr. If PH (E) is weakly convergent for E 6 (81 then it is so for E G, (B . To prove the sufficiency we now show that if Xne K, [Kn], 4 C, n 21 then there is a sub- sequence which converges weakly. It is easy to see that there exists a separable sub- space 30 of 1 and a Borel-field (Bo con- tained in ’8 generated by a denumerable number of sets { E } such that n {Kn} 6 L,(~Q.Bo, P, 360) 37 Let Z) be the field generated by { En} . 2 o evidently has only a denumerable number of sets. Now for any E E 20 “ SEW“ é £11xntldPé[xn].é c and I being reflexive there exists a subsequence n .1 such that SXnJ dP converges weakly. Since 20 has E only a denumerable number of sets we can choose a sub- sequence {:13 by Cantor's diagonalization process such that S. Xn dP converges weakly for every EE :0 . 1 E Now because P(fl ) (00 the uniform integrability of “xn |\ implies the uniform countable additivity of the i set functions an1 dP and hence by virtue of the E preceding lemma we have the weak convergence of an dP i E for all E 6 a) . Hence fan , 6‘) up i * *- 0 converges for all simple functions Y ( ) E L ”*0, . 38 DO I... me) = grog?) n x1*l\ e M Now S< xr11 , 1(a)) dP k = lim 5; x *( x <1?) i k ->°° i=1 3 n1 5f EJ % L- and the limit exists uniformly in "1" because ii N+p *( N+p * x dP {— z x I x dP EN 3 [hi )I Jflu an n n1" 1'43.1 11:J é M .fllxnfl dP dP 11-1’00 n1 K * ) = lim lim X dP . k 2: a( in “J Because an arbitrary Y*(&3 can be uniformly approximated by a countably-valued function in L°°(3E*O) 39 this proves that Xn are weakly convergent and by 1 Theorem 2.3.1 it must converge to a function xo(co) € Lfixo). This completes the proof of the theorem. Chapter 111 Strong martingales of Banach-valued r.v.'s and their mean- convergence. 1. Following Doob (1, pp. 294) we define a discrete parameter strong martingale of Banach-valued r.v.'s as follows: Definition 3.1.1 Let (Q , (B , P) be a probability space and i an arbitrary Banach space. Let I be a subset of the set of all integers and let Xt(¢d) E L1(fl,m , mi ) for all t E I . For each t E I let there be a Borel-field ‘5 t CB such that (5 C (5 whenever s “P - n. éGEHIXW) in”! E m)” 12. Also 42 because n X“ é E(" X“ [‘5 ) a.e. and hence " X“ ’4 E("Xfl‘t' 3') by Jensen's inequality. = (flxmmt dP' )1/t = E th so that “ T|| 4; 1. Actually |[ Tl| = 1 as we can show by taking X(€0) '5 a where u a“ : 1. According to the above lemma, we can associate with every martingale { Xt’ (alt, - t G I r} , a sequence of Operators Tt’ t GI defined by Tt x = E(X [ '51:) and hence mean convergence of martingales can be con- sidered from the point of view of convergence of the sequence of operators Tt . In the following section we shall make this statement precise. In theorems 3.2.1 and 3.2.2 we show that the Operators T converge to an Operator T in the strong tOpology if I. : (n2 1) or (n 5-1) respectively. 2. Theorem 3.2.1. Let { Xn, gln’ n2 1} be a martingale such that x = E 2| n71 n ( fin) — where 2(0) 5 Lyn-.43 . 9,36) 121. - x, arbitary. Then 43 lim 1 [X - ‘6] : 0 where n -€>00 n f x“° = E(z|‘3ia,) w and at :.- Borel-field generated by U 3 . n:1 n oooooooooooooo(30201) Proof: Let us define TnZ : E(Z ‘?}h) for any Zé Egg). ,IB , P, I) kZ1. By lemma 3.1.1, Tn is a linear bounded Operator mapping Lr( 9- , r8 , PJI ) into Lfdl , (in’ P, ’X) . The conclusion of the theorem then asserts that the sequence of Operators Tn converges in the strong topology to the Operator Ta: on Lk(Q, (B , P, I ) where T” (z) = Emmi.) . We shall give two different proofs of this. Our first proof applies only to the case when x is reflexive and is based on an application of a very general mean ergodic theorem, due to Eberlein, 1949, (16). This method of proving mean convergence for real-valued martingales was used by Jerison 1959 (17) in the case of martingales with index set n‘ér-1. Our second proof is elementary and is based on an application of the Banach-Steinhaus theorem and is valid for an arbitrary Banach-space iii . PrOOfI. x reflexive. We shall first state the mean ergodic theorem in the form we shall apply it. Eberlein (16) proved it more 44 generally for linear vector spaces, his theorem being a generalization of similar theorems of Yosida and Kakutani 1941, Birkhoff and Alaoglu 1940 and Day 1942. Eberlein's theorem: Let G be a semigroup of bounded linear transformations on a Banach space if . A net (Tu) of linear transformations of TI into itself is cafled a system of almost invariant integrals for G if i) for each x e x and all a(, T x 4. belongs to the closed convex hull of {sz 'Te G} ii) “ Ta“ 4 C, C independent of at . iii) for every x; x and T e G lim(fl&x-'&x)=]im(aTx-fax):C). Now, if for a given x E; , the net de has a weak cluster point I then Y, = lim T‘x in the strong topology of x . We shall apply the above theorem to the Banach space Ltd), 3;,P,I) i221 Define SnX : X - E(X‘rin) XE Lr(fl a is Fax)! n2 1 Then S S = S o o o 0(50202) m n maX(m, n) as 3.1ng = smi x - E(X I 31(1)} X - E(X'g'n) " E(X ’ E(X'En) '3!!!) 45 x - E(xfih) - E(X \3'm) E(E(x Vin) rim) + : X - E X - E X ( 1331) < Vim) E(X ) + “321n(m, n) by Th. 1.2.5 Hence SSX‘:X-EX g = m n ( ' max(m,n)) max.(m,n) so that G = (Sn, n 2 1) is a semi-group and accord- ing to lemma 3.1.1 Sn's are bounded linear operators. We shall show that the sequence of Operators (Sn’ n 2 1) themselves form a system of almost invariant integrals for G . Condition (1) is clearly satisfied. Now Sn : I .- Tn where T is as defined in 3.2.1. n “ Sn“ 6. II“ + n Tn“ = 1+ 1 = 2....(3.2.3) by lemma 3.1.1 so that S s are uniformly bounded in n norm. Also, for any m lim (SmSnS - SnX) : lim (SnSmX - SnX) n n : 0 because of (3.2.2) . Thus all the conditions (1) - (iii) in Eberlein's theorem are satisfied and we can therefore conclude that whenever SnX has a weak cluster point Y, SnX actually converges strongly to Y. 46 Now 11‘ £71, and XELrLCl, 3; , P, f) we have [Snx] 5 “Sn“ ' E X] r t 4 2°]:th because of (3.2.3). Hence { SnX, n2 1} is a bounded set in Egg, '5‘” , P, x )3 1 being reflexive so is Lyn, 3'00 , P, x ), (Th. 2.2.1) and hence every bounded set in L2(Q, a. , P, x) is (weakly compact. Therefore, { SnX } has a weak cluster point and hence according to Eberlein's theorem limdo SnX exists n in the strong topology. Of L1,(.Q , 371‘ , P, I. ). If p = 1, {SnX, n2 1} is still a weakly compact set in L1(fl,' .310 , P, x) . This is so because {H SnXu, g1n’ 1 é n $00} is a semi-martingale of real-valued r.v.'s and hence from Doob (1) Th. 3.1 (iii) pp. 311 we conclude that " SnX “ are uniformly integrable. Also S X is bounded in norm in n L1(fl , i , P, 2%) . Therefore, {SnX, n21} is weakly compact by our Th. 2.4.1. Thus, we have shown that for any Xe Lf(Q , a , P,X) 1’21 lim 3 x exists in the strong n ——)0 n tOpOlogy or in terms of operator theory, the sequence of operators Sn converge strongly. 47 Let Y : lim S X on n ‘9” n 9° (the limit on the right exists because X006 Lyfl, 3",. P. X) (lemma 3.1.1). Now E(X.o | 9n) : Xn a.e. . . . .(3.2.4) and hence for any E E 3n X dP = S dP o o o o 0(30205) SE °° EXn Also by (3.2.4) I” = 11m (1 - T ) x = x - lim xn n ——)co n n—% . . .(3.2.6) 1.6 lim [X - - Y = 0 n “'9‘ d Xn d f Hence for any E G I,- Soc.o -xn- Y“) —9 0 strongly 1n} E SOC” -Xn) dP—5—9 {11de E E or SXndP 4-) §(X¢ - Yo) dP (3.2.7) E E From (3.2.5) and (3. 2. 7) we have for every E 6 gm ch0 C11? : S(X&-Yw) (11’ E 48 n . co Hence I dP : O for all E € U fi n:1 E .0 Now U 3' n being the field that generates the Borel- n-1 field '3'“ we must have gYaOdP = O forall E63” E and hence Y” = O a.e. From (3.2.6) then it follows that lim X = X n__)‘mn 00 and this completes the first proof of the theorem. on Proof (ii) Let I; = U 3' . Because 0 n:1 n ?n C (En-+1 9 370 is a field. We shall need the following lemma: Lemma 3.2.1 The class of simple functions measurable with respect k to ($0 (i.e. functions like X(n) = Z 31 {XEfu} 1:1 «.3 8.16 I , E1 E;- 30) is dense in ”(0. 3r". PHI) 1‘21 . Proof of the lemma: Let E e (in . Be a theorem in measure theory ((4) Halmos, Th. D, pp. 56), for any. €- >O , there exists E0 E {510 such that P(E A E0) = P(E — E0) + P(Eo — E)4€ if x(eo) -_- ’XEm).a E632, , aex we) «Eco»... Eoe'3vo. P(EA Eame 49 then [X - Y1 ( £2- " x _ Yuf dP ) 1/2 H a“ ( 5“?)1/réfiall -€1/r 0 e being arbitrary it's clear that we can choose Y(&>) measurable with respect to 30 and as close to X(fl) in Lf( x) norm as we please. Hence any simple function in L£(Q , 3i. , 13,1) can be approximated by simple functions measurable with respect, to '310. Since the simple functions are dense in Lr(1 ), ((2), Dunford :3: Schwartz, pp. 125), we can approximate an arbitrary function in LP( 1) by simple functions measurable with respect to 30 as closely as we wish. This proves the lemma. Consider now the following sequence of mappings from the Banach space L2(Q , 30° , 13,1 ) to itself Tn: 3( Q9 3'» 9 139%) —‘—'? Lt(fls 19 Psi) TnX : E(X l 5n) Now 1/r E j ) P 83p TnX P : 83p “ TnX n . dP é [X]? since " TnX" t 5 {Eu mm)? 5- E< «x» ’13:) .0 -01 50 Hence the set (TnX, n 2 1) is bounded for each x (a Lt((2.,‘?&, , P;;SE) . If x10» = ‘7(F(an.a where ae K , F 5 ‘50 then since for some N, Fegnvuflwe have TnX : X a.e. for n2 N . Hence n3; [Tnx'ah = O . . . . (3.2.8) and so (3.2.8) is true for all simple functions X(Co) measurable with respect to 3' o . Since SUCh functions are dense in Lr(Q , a. , Pix) according to lemma 3.2.1 we have by the Banach-Steinhaus Theorem (3) that lim TnX : TX -. . . . (3.2.9) n—900 exists for all X5 Lyn . ‘3... . hi) and moreover that T is a bounded linear operator. For X(€O)'s which are simple functions measurable with respect to (3‘ we have TX = X . 0 Such functions being dense in Lyn : 3.0. P, i ) we can obtain, given an arbitrary X, a sequence Xn of them such that lim [: X. - X:‘ n-—)oO n i ll 0 T being continuous we have TX : lim T : 11m : X ..(3.2.K» n—aco Kn n——)OO Xn 51 Hence, we have proved ((3.2.9) and (32.10)) that E(X | 3 ) converges in L (.Q , 3.. , RI) n ! norm to X for every X e Lt(fl' a. , P, 1 ) . The theorem then follows by taking X:E(Z\?ao). Theorem 3.2.2 Let (Xn(6>), ‘5, né - 1 ) be a martingale n with "in 3‘31““) , x_1 1 . Then X (0) converges in Lr(1) norm to n X _ (m) i.e. lim L X - X_n] : O n ——)¢O '00 f where X“.D (0) : E(X_1(0) ' 3- an) a.e. and 7? ’3 '3'.“ = n = 1 -n Proof: We shall againrresent two different proofs; the first proof uses Eberlein's mean ergodic theorem with the additional assumption that x is reflexive and the second prof is based on an application of Banach-Steinhaus theorem. Proof (1) iii reflexive: Define. the bounded linear operators Tr1 on Lr( Q , $1, P, i ) to itself as follows: 52 TnX = E(X‘g_n) x6 114.0, 34, P,§),n21. Then II TnX“ 5 E(“XII | (in) a.e. and by Jensen's inequality IITnXII’ é E("xui’l 3-..) [Ts], = ( Llama)” 4 Ex], so that TnXe LED 9 $1. P. I) ( .}_nC 34) Hence and )ITn H = 1 for all n Also Tm 3 Tn : Tmax.(m, n) Hence G : (Tn, n 2 1) is a semigroup of bounded linear Operators for which. (T3, r1231) itself is a system of n almost invariant integrals for G. Hence by Eberlein's mean ergodic theorem (see Th. 3.2.1) TnX_1 : X-n“”) goes to a limit in the norm of L’(..O. , ‘3'”, P, x ) whenever it has a weak cluster point. Now Lr(fl, (34, P, x) for p )1 is reflexive (Th. 2.2.1) and hence every bounded set in it has a weak cluster point. But [xm]? 4 [x_ L so that the set {TnX_1} or { X41} is bounded and this, in con- Junction with the previous comment, proves the assertion of the theorem when p >1. When p : 1 we notice that {" X_n“, 31w n21} 53 is a semi-martingale and hence by Th. 3.1 (iii), pp. 311, Doob (1) we conclude that “X n“ are unifomly integrable. Also {X n} is a bounded set in L1( I ) and so by our theorem 2.4.2 {X 1 is weakly compact. Hence X n converges strongly in L1(z ). We shall now show that lim X__n = X_.o . n —)w Let lim X n = Y. Then E(Y ‘34” = Y for n ——)ao " , all m 2 1. i.e. Y is measurable with respect to $411 for any 111 2 1 and hence measurable with respect to “inf Also for any A 6 3 .0. 5 Y dP : lim X_n dP = X_1dP n —-)OO A because X = E(X_1‘ -n)° This proves that Y : X .0 a.e Proof (ii) Define Tn: 1214.0, 3—1’ P, I) a Lr(Q! 9-19 P’fi) TnX = E(X ‘ 3%) n2 1 T are bounded linear operators (lemma 3.1.1) such that n supfwndr 4 [at .....(3.2.11) n Also if x(co) : «SUBLa , a e x , E e (5” then TnX = P(E )'3'_n).a Now P(E Ig-n) converges to P(E I 3"») in L P i.e. 54 '/ k t n 21;” J‘HE FTC”) - P(E "in” dP): o . . . . . (3.2.12) This follows either by applying Doob's Th. 4.2, pp. 328, (1), or by considering P(E |g'_n) as a real-valued martingale and then applying Proof (1) which is applicable since the real numbers firm a reflexive Banach-space. It follows from (3.2.12) that 11‘ xm) = ’XEw).a then TnX converges in Lr(1). Hence TnX converges for all simple functions .XGD) which form an everywhere dense set in Lt(SE.). This and (3.2.11) enables us to conclude by virtue of Banach-Steinhaus theorem (3) that lim TnX = TX n -§@° in Lt(1) for all X 6 L“!) where T is a bounded linear operator. This proves Th. 3.2.2 for a general Banach-space If . g3. In this section we shall prove mean convergence theorems for arbitrary martingales of r.v.'s taking values in a reflexive Banach-space. We shall need the following lemma: Lemma 3.3.1 Let Tn’ n = 1, 2, ... and T be bounded linear operators mapping the Banach space E into itself and C' such that 1) lim Tnx = Tx for all x51 n -€’ao and ii) Tm“ Tn : Tmin.(m,n) 55 Let xne x such that there is a subsequence x converging weakly to x and also that nk «3 T x : x . n n+1 n Then lim xn : x” strongly. n ->co Proof: From the conditions of the lemma we have T X : . . . . . 0 (3.2.13) m n Xmin(m, n) Also menk a) ) mea as k ——-)°O for any m . By (3.2.13) men : xIn for large k so that k meco : xIn Now by condition (1') of the lemma med —S—) wa so that it follows that 11m x = x strongly. m——)aO m ‘0 Theorem 3.3.1 Let f be a reflexive Banach space and let ,i Xn, ‘E'n, n2 1} be a x-valued martingale such that Xne LED-KB, Kg) n21, p)1 and [Xn]P Lp(Q, ‘51»,35) 56 -_-. E(X (.35) X Si::: Borel-field generated by m U 3" n=1 n We have Tm’Tn : Tmin.(m, n) and by Th. 3.2.1 lim TnX : X for all n.—4;ao XELflQ, 3&9 P9?)- Also Tnxn+1 : Xn because {)8}, 3n, n? 1’} is a martingale. [xn1p 5 ll Tn" '[XnH-Jp : [Xn+1]p < C so that {X113 is a bounded set in LP(Q’ goo P, CE) which being reflexive (°.‘ 1 is reflexive) Hence (Th. 2.2.1) the set .{X n‘3 is weakly compact i. e. there is a subsequence .XO‘S converging weakly to some element, say Xco e Lp(Q ,3“, P, f). Thus Tn's and Xn's satisfy all the con- ditions of lemma 3.3.1 which therefore guarantees the assertion in Th. 3.3.1. Theorem 3.3.2 Let i be a reflexive Banach space and let { Xn, $1.1, n z 1} be a 1' -valued martingale such that xn e L1(Q,’B,P,f£) Suppose that 'lIXn||'s are uniformly integrable. 57 Then there exists x” g L1(fl , ’5 , P, 1 ) such that 11‘“ [Xn' «0.11: O n—-)ao Proof: We define Tn's as in the previous proof and then the previous arguments would prove the assertion in this theorem if we could show that {Kn} is weakly compact in L1( x). This we do by first showing that [Xn]1 < K , K independent of n . Because E(Xn+1\ an) = Xn a.e. an+1dP = Xn dP for all E e fin. E E Hence S‘Xn dP converges strongly to a limit for every E ac Ee U n. Now let /‘§(E) =jig1d1? E632, n:1 E /:(E) is uniformly countably additive on 52: i.e. if Enc:En-1 and I EF; (13 En e: 00 ’ A:} En : X then lim (E ) : O uniformly in m n -—)OO m n This is so because "/1: Min)" _ £3“ x1m n dP n and because )‘XnGD)II are uniformly integrable. 09 Since U 3n generates the Borel-field 3:0 , n:1 it follows from Lemma 8 pp. 292 (2, Dunford & Schwartz) 58 that Km) converges strongly for all E e ‘35, . #1 Hence, also for any x*e x lim x*(f X n dP) n —-)¢O E exists so that 11 co,Y*a dP n mj(xn() ()) converges for all Y*(€o) é Lw( x‘”) which are simple. Hence, it can be shown that as j ), Y (an) dP 3|; ’1‘ converges for all Y (0)6 Lao ( x ). In other words, Xn(a>) is weakly convergent in L1( 1%) and hence bounded ((3), pp. 36). { Xn(o) } being a bounded set such that [I Xn(co)||'s are uniformly integrable it follows from Th. 2.4.1 that {391(0)} is weakly compact. This terminates the proof of the theorem. Chapter IV Almost everywhere convergence of Banach-valued strong martingales. £1. In this chapter we study the almost everywhere con- vergence of certain special types of martingales, namely the ones generated by taking repeated conditional expecta- tions of a fixed r.v., and other cases which can be reduced to this case. Our proofs are quite different from the ones used by Doob (1) in the classical real or complex-valued cases. We use a theorem originally due to Banach (1926, 12) and a generalized version of which is in (2, Dunford & Schwarz, pp. 332, Th. 3). As pointed out in the foot-note of a paper by (Schwarz & Beck, 1957, 13), the theorem can be extended to Banach-valued functions without any change in proof. We shall state the theorem in a slightly restricted form in which we shall apply it here: Let Tn be a sequence of continuous linear Operators on a Banach space "3 to L1(fl , m, P, f) such that i) sup “ TnY(fl)“1 3% be any martingale taking values in any Banach space Then 13(6): sup " Xn(0)" : +00) : 0 if n E(“ Xn" )< 0, independent of n . Proof of the lemma: Let A : (Co: sup H X1100)" = +00) n AM: (GO sup |1Xn(w)l| 7 M) n Then m A = M21 AM and AM :3 AM+1 w 7 Now AM : 1-? Bi where Bi = (w: llx1(w)fl> M, 1| X10011 éM. ||X2(co)|| éM, . . . "Xi_1(‘°)" 4. M) Bi's are disjoint and B1 5 (3‘1 . Since Bi 5 3'1 and [I X1(‘-’)" 5 E( “KN“ \ 91) for N:? i , we have In )LNfldPZ jflxiudP > M p031) B1 B1 Hence N x d]? = N N3." N" 14E? BjuxNfldP > M 5 P(Bi) i U B 1:1 1 62 so that N 1 1 Q P< 1,31% 5, S 11 mm 4 mjllxNNdP 4M N U B i=1 1 Hence taking N ———fi>°° C P(AM) < M Hence lim P(AM) : O M ..9eo But, lim P(AM) = P(A) M -—)Hfi and so P(A) : O This completes the proof of the lemma. If 2(0) : «E(w).a where a e 1 , E 6 ‘3; then we have TnZ(¢-3) 3' Z(€O) for n 2 N. Hence 0 lim TnZ(€0) exists for such 2(0) r1-—€?¢O and hence for simple 2(0) measurable with respect to «3'0 . Now let us apply the theorem mentioned in § 1 IE5 ‘= IH('(): €£Lp ::Ps 3E ) EUZIEFn) According to lemma 3.2.1, the simple 2(0) measurable with respect to 3’0 are dense in L1(.()., '1, P, 1 ) and hence, all the conditions stipulated in the theorem with TnZGD) in § 1 are valid. Thus, we can conclude that lim E(X lfifin) exists a.e. strongly for every n -—9mo 63 2(0) e L1(-Q. goo max). For any 2(0) 6 L1( fl , (B , P, X) we consider x00 (0) = E(zl‘i)eL1(Q.1P.1 ) and as . xn = E(z|‘}rvl) = E( Xalfin) it follows that lim X (a) n -—)‘° n exists a.e. for any Z 6 L1(Q , (a , P,%). To show that the limit is indeed X410 (0) we simply observe that )%(0) converges in mean to X a) by ,( Th. 3.2.1 and that a.e. convergence and conver- gence in mean are compatible. This concludes the proof of the theorem. Theorem 4.2.2 Let {Xn(fl), gm, n 2 1} be any x -valued martingale where z is a reflexive Banach space. Let n Xnm)“ , n 2 1 be uniformly integrable. Then there exists a } -valued r.v. X00 (0) such that 1 lim X (0) z: X (GO) a.e. n 400 n 00 Proof: According to Th. 3.3.2, there exists Xao("’) E L1(fl, at!“ , P, 1) such that lim —-) Hence lim [- X - X ] : O n a” n & 1 n a andr = xde AGE, ”(4.25) A A As E(Xn+k'g'n) = Xn a.e. kzo we have Making k ——-)00 we have because of (4.2.3) fxndP : [X‘adP for any B e gn . B B This means that “Kati (yin) = Xn a.e. n21 From the preceding theorem, then, we can conclude that n31; Xmas) : Xeo(‘°) a.e. Our next theorem is about the almost everywhere convergence of martingales with decreasing index set. Theorem 4.2.3 Let { Xn, (in, n _l_ -1 } be any x -valued martingale where the Banach space x is arbitrary. Then lim X = X to a.e. n w nko) _m< ) where X (”)=E(X1(w)\31 ). E = A 3n '& - "M w n 4- -1 Proof: We notice firstly that X_n : E(X_1‘$_n) a.e. Define T as the continuous linear Operator from n L1( a , 51: P: x ) to itself, given by TnX : E(X rim) . The proof will be completed by showing that for every XE L1(Q , 3‘1, Rx), Tn X converges a.e. That the limit is the prescribed one follows by 65 noticing that it is also the limit we get from the mean convergence of TnX (Th. 3.2.2). If X(“) = (x (0)0a 9 86 x 9 E e ‘31] E Because n Sm” P(E ‘3-n) = HE‘E-lw) a.e. (see Doob, 1) lim T X = P E I .a : E X I a.e. 11...)» n ( 3E") < EL”) Hence lim TnX exists a.e. for any simple n -—)m0 function x(a) e L1(Q. P751. P. 1%) Also 83p H TnX((-1)“ : Strip" E(XI £31K» a.e. as one can show by a proof similar to lemma 4.2.1. Now by an application of the theorem mentioned in §1 we can conclude that lim TnX exists a.e. for every ‘1-€?°° X6 L1(S)_g 3L1, 15%)- This finishes our proof. 33. In this section we shall prove an almost-every- where convergence theorem by using the idea of optional stopping (Doob, pp. 300, 1). Let man) be a random-variable whose finite values 66 are positive integers and which may be +00 with positive probability. Let (Xn, 9n: n 2 1) be a E -valued martingale and let {a}: M“) = k }6 u We shall define the random variables Xn(0) n 21 as follows u X369) = ij) ”9&0: .15, 111(0)} e Xm(0)(“’) cafe): 3) 111(0)} Lemma 4 3 1 u {Xn’ n’ n21} is a x -valued martingale. Proof: Because {a}: m(0) = k}€ 3k we have {a}: m(0) >n }€31n-1 and Hence {co m(0)< n}e‘3' _1 U (a) = (w) (o) n" “(0) )91 Xn a “2)1’1} + 1E1Xk(0){m(u) =k} i s gn-measurable . \1 t: We shall now prove that E(X ‘3' : X a.e. n+1 n) Let A 6 3'; . Then U V v Xn+1 dP = Xn+1 dP + X +§P A A m((.:) 4 n} An m(0) > n} 4' ti \1 Now on { m(o) _ n ’ Xn+1 : Xn and 67 V {111(0) > n} e 3n ; also on the latter set Xn n and hence by martingale prOperty U U u SKIN" dP : 5' Xn dP + J. Xn dP A II N A An{m(¢0) g n} “{mM) 7 n} u A This proves the lemma. Theorem 4.3.1 Let {Xn’ 3n, n 213 be a % -valued martingale where 3 is a reflexive Banach space. Let E( sup “ X510) - X153; 1' )4 +w. XOR») 50 n > 0 Then lim xn(co) exists whenever n —-)°° COG-{€93 sup ”Xn(0) " < +00} n Proof: Let M)O be any positive integer and define mm) = n 06{||X1(0)||§M.. . . 11Xn_,(w)ll 5M. 11 maul) M} = 00 636 {8:13 "Xn(w) \I< + 00} Of course {m(Q) : n }ec$n . Let X , M (U) be defined as follows 68 u Xn. M (a) = Xnm) we{m(m)> n3 HG”) G3€ {imflb)¢L n} Let Y(O) znsgpO u Xn(0) - Xn_1(9) ” 9 Then I. Xn,M(u)“ é M + Y(CO) , E(Y)< + co and so u A H Xn, Mm)“ “ Z 1 are uniformly integrable. According to lemma 4.3.1 {31 Vinny} n, n is a martingale and so Th. 4.2.2 allows us to conclude that U lim X , (w) n a” n, 4": exists almost everywhere. U Since xr1 M(c>) = xnm) if jsgp1 u xjuo)" 5M the theorem is proved. Corollary. Let f X , $11, n 2 1} be a if -valued n martingale, ‘BE reflexive, such that E(“Xnu )< C (independent of n) E( sup “ Xn- Xn-1“) < +00 n2: 0 Then 11111 X (0-1) exists a.e. n —-)°O n Proof: By lemma 4.2.1 P(‘O: sup " Xn(0)"< + ca) : 1 69 and this fact combined with Th. 4.3.1 implies the statement in the corollary. Analogues of other theorems for real or complex- valued martingales can be proved as in Th. 4.3.1 for reflexive Banach spaces. We omit them here for brevity. $4. In this section we should like to point out what can be done by direct applications of convergence theorems about scalar valued martingales (see Doob, 1). The fundamental idea here is simply that if {Xn’q‘n’ n 21} is any a -valued martingale then { x*(Xn("))' '55, n21} * as” is a complex-valued martingale for any x G . Theorem 4.4.1 Let {Xn’ an, n 21} be a x-valued martingale where x is a reflexive Banach space. If E(Il Xnu)< C independent of n, then there exists a Bochner-integrable r.v. X” (0) such that Xn(6)) converges to - x.°(u) a.e. weakly. Proof: Because each Xn(fl) is Bochner integrable, there is a separable subspace 10 C i such that £160) 6 $0, n21 except for we N, P(N) : O. ** % being reflexive so is x 0' Also, Io : E0 = * * (x2) and hence fife is also separable. Let x )1! {x E x be a denumerable set of elements 1 * o * dense in x . Now for each x1 {$.23an gm n21} 70 is a complex-valued martingale such that E<|x§a° 1 xn exists for all 1 except for 06 M. P(M) : O. a: If Let x 6 10 be an arbitrary element and let * * x:L ? x k Then if caé M |x*(Xn(°)) - x*1 4 |x*(xn(e)) - x: (xnwnl +| x*) as follows 11 6 E m ) W =2 ,_2_§__,_ in “(a -t-J- : O elsewhere. It can be easily seen that {Xn’ q'n, n 21} is a martingale and that “Xn(¢o)" E 1 a.e. ll d n21 E(ll M”) II) E( w - 6) = 1 X = O nsépo“ xn< ) xn_1< >ll> 0 But if w # p/2q then Xnaa) does not go to any limit either weakly or strongly. Actually no subsequence Xn (0) converges weakly or strongly if u i p/2q . Hence Xn(0) does not converge in L1( x) - mean either. N. B. _ The function 61(t) from the unit interval to L1 was given by Clarkson (5) as an example of a Banach-valued absolutely continuous function having derivative almost nowhere. Our construction of Xn(w) is patterned after Doob's (1) method of applying martingale theory to the theory of derivatives. Chapter V Some applications of the general theory. §1. In this chapter we shall show how martingale theory can be brought to bear upon some classical problems. We shall not aim at exhausting all possible applications of our theory of Banach-valued martingales, but shall rather indicate how our theorems can tackle the extensions to Banach spaces of some of the problems which Doob (1) has considered in the complex-valued case. § 2. In this section we consider two different types of strong law of large numbers for a sequence Yn(¢d), n = 1, 2, .... of independent, identically distributed r.v.'s taking values in an arbitrary Banach space 3. . The results are stated in theorems 5.2.1 and 5.2.2; they generalize Mourier's (10) results which were proved for separable and reflexive spaces. However, Mourier's results are more general in the sense that they concern arbitrary stationary sequences. Theorem 5.2.1 Let Yn(€o), n = 1, 2, . . . . be a sequence of independent, identically distributed r.v.'s taking values in an arbitrary Banach space a and let Y1(G) be Bochner-integrable. 73 74 If n sum) = 2: Y1(“) 1:1 5 a then lim n( ) : E(Y1(U)) a.e. n ——€>OO n N. B. Two a -valued r.v.'s 21(6), 22(0) will be said to independent if for any two Borel sets B1, B2 of SE P(w: 21m) 5 B1, 22(0) 5 B2) : P(w: Z1(0)€ B1) P(u: Z20») 6 B2) They will be said to be identically distributed if for any Borel set B of :!E P(GO: Z1(o)é B) : P(Cd: 22(03) 6 B) 1 ed and one of them, namely Y Proof: Because the X s are identically distribut- 1 is Bochner-integrable, so are the rest. Hence we can consider their conditional expectations relative to any Borel-fields. According to Theorem 4.2.3 lim E(Y S. S n _;’°°’ 1‘ n’ exists a.e. (in the following, the symbol n+1’ ...) — -w E(z(¢o) | ztm), te T) shall stand for a conditional expectation of the Bochner- integrable r.v. Z(€o) relative to the smallest Borel- field with respect to which the family of r.v.'s 75 Zt(u), t e T are measurable .) ..) Now E(Y1|Sng Sn+1y °'°° ) = E(Y113n"yn+1’ Yfl+2’ : E(Y1‘Sn) since the Yi's are mutually independent. Hence lim E(Y |s ) = lim E(Y s , s ,...) n §°° 1 n n a“ 11:1 n+1 : X '00 exists a.e. Also, as Y 's are identically distributed E(Y1‘Sn) = E(YJ‘SH) a.e. 1_4_J§._n so that l n — 17' E(M,|sn) - n 51 ~(Y313n) a.e. 1 s _ .. — .9 _ n E(Sn)§f _ 11 a.e. s (so) . Heme 11m 2 = lim E(Y1‘Sn) = x_°° n -—)00 n ‘;?‘° a.e. exists. That x_w(o) = E(Y1(0)) a.e. follows from the fact that X : constant -00 : E(X ) a.e. -oo because of the Zero-one law and that is a martingale. This completes the proof of Theorem 5.2.1. Theorem 5.2.2 Let Yn(¢0)', Sn(0) be as in Th. 5.2.1. If Y1(0)€Lp(n, 8,1393) 1épOD 11 .f Proof: According to Theorem 3.2.2, there is X 6L(fl,%‘,P,%) suchthat "fl p n iaLEWASn, an“. ) - >811? = o The conclusion of this theorem then follows by proceeding exactly as in the previous proof. §,3. In this section we shall consider the problem of existence of derivatives with respect to nets of a countably-additive Banach-valued set function defined on an arbitrary probability space. It would be clear from our considerations that similar results can be proved for ab-finite measure spaces. We limit ourselves only to the case when the set functions take values in a reflexive Banach space. Examples due to Bochner (7) and Clarkson (5) clearly 77 indicate that some restrictions on the Banach space are necessary. We shall first state a lemma: Lemma 533.1 Let (.0- ,(B , P) be a probability space and let gvn} , n 21 be a sequence of Borel-fields such that gnc (3 for n71 and each 3n is generated by a finite or denumerable number of disjoint sets { Mgn) , 3 Z 1} i.e. Efih is the smallest Borel-field containing {M3130 , 1:2 1} . Furthermore, for any n and 3 let there be a k 2 1 such that n+1 (n) M} ) C’ Mk Let ? (-) be a countably-additive x-valued set function defined for sets in 27 ’3. n=1 and let (I be a reflexive Banach space. Let Xn(0) be a sequence of I-valued r.v.'s defined as follows: awn» ' (n) (n) xnm) W if menJ , P(MJ ))o = 0 otherwise. Then there is a r.v. x(¢o) 6. L1( Q , go", . RX) such that 78 97(A) = SXKO) dP (where a is A an the Borel-field generated by U G}; ) n:1 for all m A E U qn if and only if the real- n:1 valued r.v.'s " Xn03)1) are uniformly integrable. Proof: It is clear that 5n C €511“ for n 21 and Xnab) is measurable with respect to 'EE. If "Xian-3) I) 's are uniformly integrable, then Xn09)'s individually are Bochner-integrable and so it is meaningful to talk about their conditional expectations. {Kn’ '5‘... n21) forms a martingale. According to our Theorems 4.2.2 and Also 3.3.2, there exists a r.v. X(0) é L1(Q, i, P, f ) such that lim XnGD) : X63) a.e. n-—J)OD and n figmo [:Xh - Xi11 : O and E(X [‘11) - xn a.e. Hence 79 “"3" xcr for AeU n n:1 This proves that 9m A Conversely, i f ?(A)=S.xcP forA€I°Ja 3n A . n:1 then E(X13h) : Xn a.e. and hence " Xn ‘| 's are uniformly integrable. N. B. A sequence of partitions {1111311)} of I)— as in the lemma is called a net. The function X(w) is said to be the derivative of q with respect to the probability measure P, relative to the net { Mg“) } . For the formulation of our next theorem we need the concept of "total variation" of a x -valued set function 9 defined on a field '3' . We define the set function 199(A) , A e g! , which we shall call the total variation of 9 on A as follows: 195%) = {sup {.51 I) 9 (A1) 11} where the supremum is taken over all finite disjoint A. sequence A1 of sets in 3:! such that A1 C Clearly n q (A) “ é ”9(A) If g is countably additive on , then 199M) 80 is also countably additive on g . Theorem 553:1 Let ( Q. , (B , P) be a probability space and let 9n, 1, g , and ’X be as in lemma 5.3.1. Then 9(A):)X(w)dP Asa: 3:651, n: A where X(co)€ L1(Q.3;.o.1=.%) w 11" and only if «99(A) on ‘50 = U n n:1 is finite and absolutely continuous with respect to P i.e. for any 6 7 0 there is S 7 0 such that whenever P(A)4S' , and A 5‘32) . Sufficiency: Proof: If ”9(A) is a finite, non-negative, countably additive measure on '3‘0 which is a field, gu— then it has an unique extension 4% to the Borel- field i» generated by go . It follows from simple considerations that :59 on 3” is absolutely continuous with respect to P if 199 is absolutely continuous with respect to P on 310 . According to the Radon-Nikodym theorem, there is a non-negative function 1(a) measurable with respect to '31” such that 81 69M) = S Y(0) dP As 1 . A Define Xn@b) , Ynab) as in lemma 5.2.1 by means of 9 and 7139 respectively. '— Clearly I|Xn(63)" éYn(€0) a.e. n 2 1 4%? 'being an integral, it follows from lemma 5.2.1 that Yn's are uniformly integrable. Hence, IIXn" 's are uniformly integrable and this implies according to lemma 5.2.1 that g has the integral form as stated in the theorem. Necessity: If g (A) = f X(0) dP then A 199M) 4 I “ xm) 1| dP A and this immediately proves that ”,(A) is finite and absolutely continuous with respect to P on 10. 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