Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/weaksequentialco1489yann 
 

 ST 
 
 BEB 
 
 FACULTY WORKING 
 PAPER NO. 1489 
 
 Weak Sequential Convergence in L p (/x.X) 
 
 Nicholas C. Yannelis 
 
 College of Commerce and Business Administration 
 Bureau of Economic and Business Research 
 University of Illinois, Urbana-Champaign 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 1489 
 
 College of Commerce and Business Administration 
 
 University of Illinois at Urbana- Champaign 
 
 September 1988 
 
 WEAK SEQUENTIAL CONVERGENCE IN L (/i,X) 
 
 P 
 
 Nicholas C. Yannelis, Associate Professor 
 Department of Economics 
 
 I would like to thank a careful and competent referee for 
 useful comments and suggestions as well as Erik Balder. Also 
 I wish to acknowledge several helpful discussions with M. Ali 
 Khan and Aldo Rustichini. Needless to say, I am responsible 
 for any remaining shortcomings . 
 
WEAK SEQUENTIAL CONVERGENCE IN L (m,X) 
 
 by 
 
 Nicholas C. Yannelis 
 
 Department of Economics 
 
 University of Illinois, 
 
 Champaign, IL 61820 
 
 Abstract : We provide some new results on the weak convergence of sequences or 
 nets lying in L ((T,2,/z), X) = L (^,X), 1 < p < °o, i.e., the space of 
 equivalence classes of X-valued (X is a Banach space) Bochner integrable 
 functions on the finite measure space (T,2,/i). Our theorems generalize in 
 several directions recent results on weak sequential convergence in L..(/i,X) 
 obtained by Khan-Majumdar [12], and Artstein [2], and they can be used to obtain 
 dominated convergence results for the Aumann integral. Our results have useful 
 applications in Economics. 
 
 * I would like to thank a careful and competent referee for useful comments and 
 suggestions as well as Erik Balder. Also, I wish to acknowledge several helpful 
 discussions with M. Ali Khan and Aldo Rustichini. Needless to say, I am 
 responsible for any remaining shortcomings. 
 
1 . INTRODUCTION 
 The purpose of this paper is to prove some results on the weak convergence of 
 sequences or nets lying in L (^,X), 1 < p < <», i.e., the space of equivalence 
 classes of X-valued (X is a Banach space) Bochner integrable functions x: T -» X 
 on a finite measure space (T,S,/x). In particular, the main theorem of the paper 
 asserts that: 
 
 If X is a separable Banach space, (T,E,/x) is a finite positive measure 
 space, and {f : A e A) is a net in L (/i,X), 1 < p < <=°, such that, f converges 
 
 A p A 
 
 weakly to f € L (tf.X) , and for all A e A, f x (t) 6 F(t) u-a.e. , where F : T - 2 X 
 P A 
 
 is a weakly compact, integrably bounded, convex, nonempty valued correspondence. 
 
 Then we can extract a sequence (f : n=l , 2 , . . . } from the net (f : A e A} such 
 
 n 
 
 that f converges weakly to f and for almost all t in T , f(t) is an element of 
 n 
 
 the closed convex hull of the weak limit superior of the sequence f (t), i.e., 
 
 A 
 
 n 
 
 f(t) G con w-Ls{f (t)} /i-a.e. 
 
 A 
 
 n 
 
 The above theorem generalizes in several directions a recent result of 
 Khan-Majumdar [12], which in turn is an extension of a theorem of Artstein [2]. 
 Moreover, versions of the above theorem can be used to prove 
 
 Lebesgue-Aumann- type dominated convergence results either for the set of all 
 integral selections of a correspondence or for the integral of a correspondence 
 The latter results extend the previous dominated convergence theorems for the 
 integral of a correspondence obtained by Aumann [3], Pucci-Vitillaro [16], and 
 Yannelis [22]. Our results have useful applications in Economics and Game 
 Theory, (see for instance Khan-Yannelis [13], Khan [14, 15] and Yannelis [21]). 
 
The paper is organized as follows: Section 2 contains notation and 
 definitions. In Section 3 the main results of the paper are stated, and finally 
 the proofs of all the results are collected in Sections 4 and 5. 
 
 2. NOTATION AND DEFINITIONS 
 
 2.1 Notation 
 
 2 denotes the set of all nonempty subsets of the set A. 
 
 <j> denotes the empty set. 
 
 dist denotes distance. 
 
 R denotes the set of real numbers. 
 
 R denotes the -2-fold Cartesian product of R. 
 
 If A is a subset of a Banach space, ciA denotes the norm closure of A, and 
 
 con A denotes the closed convex hull of A. 
 
 •k 
 
 If X is a linear topological space, its dual is the space X of all 
 continuous linear functionals on X, and if p e X and x e X the value of p at x 
 is denoted by <p,x>. 
 
 If {F : n=l , 2 , . . } is a sequence of nonempty subsets of a Banach space X, 
 
 we will denote by w-LsF and s-LiF the set of its weak limit superior and 
 J n n r 
 
 strong limit inferior points respectively, i.e., 
 
 w-LsF = (x <E X : x = w-limx , x E F , k=l , 2 , .' . } 
 
 k~ "k \ \ 
 
 s-LiF = {x e X : x = s-limx , x e F , n— 1.2,..}. 
 n n n n 
 
 2 . 2 Definitions 
 
 Let (T,S,/i) be a finite measure space and X be a separable Banach space 
 
The correspondence cp : T -» 2 is said to have a measurable graph if the set 
 G ={(t,x) G T x X : x G <p(t)} belongs to £ ® /3(X) , where 0(X) denotes the Borel 
 a-algebra on X and® denotes product a-algebra. The correspondence <p:T -» 2 is 
 said to be lower measurable if for every open subset V of X, the set {t e T : 
 <p(t) n V * <i>) belongs to 2. It is a standard result (see Himmelberg [10, p. 47]) 
 that if <p( • ) has a measurable graph, then cp( • ) is lower measurable, and if <p( • ) 
 is closed valued and lower measurable then <p( • ) has a measurable graph. 
 Moreover, if T is a complete finite measure space and cp( • ) has a measurable 
 graph and it is nonempty valued, then there exists a measurable selection for 
 <p( • ) i i.e., there exists a measurable function f : T -* X such that f(t) e <p(t) 
 H-a.e. (see [10, Theorem 5.2, p. 60]). 
 
 Following Diestel-Uhl [7] we define the notion of a Bochner integrable 
 function. Let (T,S,/i) be a finite measure space and X be a Banach space. A 
 
 function f : T -»■ X is called simple if there exist y 1 ,y_ ,y in X and 
 
 n 
 
 a ,a , . . . ,q in S such that f = 2 y. y , where y (t) =1 if t G a. and v (t) = 
 1 2 n . 1 y i A a. A a. l A a. 
 
 1=1 11 l 
 
 if t € a. . A function f : T -► X is said to be u- measurable if there exists a 
 l r 
 
 sequence of simple functions f : T -»• X such that lim I f (t) - f(t) I = for 
 
 n rt-Ko » n " 
 
 almost all t G T. A /i-measurable function f : T -» X is said to be Bochner 
 integrable if there exists a sequence of simple functions {f : n=l , 2 . . . } such 
 
 that 
 
 Jig J"t I! f n (t) ■ f(t) 1 d/i(t) = °- 
 
 In this case we define for each E G S the integral to be J f(t)d/x(t) - 
 
 lim f_f (t)d/i(t). It can be shown ( see Diestel-Uhl [ 7 , Theorem 2, p. 45]) that, 
 n-»oo J En 
 
 if f : T -»■ X is a /z-measurable function then f is Bochner integrable if and only 
 if J* || f(t) dji(t) < <». For 1 < p < oo, we denote by L (p.X) the space of 
 
equivalence classes of X-valued Bochner integrable functions x : T -» X normed by 
 
 1 
 
 l|x|| p =[; T ||x(t)|| p d / i(t)]p. 
 
 As was noted in Diestel-Uhl [7, p. 50], it can be easily shown that normed by 
 the functional II • II above, L (m,X) becomes a Banach space. 
 
 ii lip p 
 
 A Banach space X has the Radon-Nikodym Property with respect to the measure 
 
 space (T,2,/i) if for each ju-continuous vector measure G : S -> X of bounded 
 
 variation there exists g e L (/x,X) such that G(E) = f g(t)d/x(t) for all E G S. 
 
 A Banach space X has the Radon-Nikodym Property (RNP) if X has the RNP with 
 
 respect to every finite measure space. Recall now (see Diestel-Uhl [7 , Theorem 
 
 1, p. 98]) that if (T,E,/i) is a finite measure space 1 < p < «, and X is a 
 
 Banach space, then X has the RNP if and only if (L (u,X)) = L (u,X ) where 
 
 P q 
 
 - + - = 1. For 1 < p < oo denote by S the set of all selections of the 
 p q r J tp 
 
 X 
 correspondence cp : T -» 2 that belong to the space L (ju,X), i.e., 
 
 S P = {x G L O.X) : x(t) G <p(t) /i-a.e.}. 
 
 We will also consider the set S = {x G L-(/i,X) : x(t) G <p(t) u-a.e. } ,i.e. , S 
 
 is the set of all integrable selections of q>( ■ ) . Using the above set and 
 
 y 
 following Aumann [3] we can define the integral of the correspondence cp : T -* 2 
 
 as follows : 
 
 J T <p(t)d/i(t) = {/ T x(t)d/i(t): x G S^; 
 
 In the sequel we will denote the above integral by ftp. Recall that the 
 
 v 
 
 correspondence <p : T -» 2 is said to be integrably bounded if there exists a map 
 
 g G L , (m.R) such that sup{||x||:x G <p(t)) < g(t) /^-a.e. Furthermore, if T is a 
 complete finite measure space, X is a separable Banach space and (p : T ■* 2 is 
 
an integrably bounded nonempty valued correspondence having a measurable graph, 
 
 by virtue of the measurable selection theorem we can conclude that S is 
 J <P 
 
 nonempty and so JV is nonempty as well. Let now (F :n=l,2, . . . } be a sequence of 
 
 nonempty subsets of a Banach space X. We will say this F converges in F 
 
 (written as F -» F) if and only if s-LiF = w-LsF = F. 
 n J n n 
 
 With all these preliminaries now out of the way we are ready to state our 
 main results. 
 
 3. THE MAIN THEOREMS 
 
 We begin by stating the following result on weak sequential convergence in 
 L (/x,X) , 1 < p < «. 
 
 Theorem 3.1 : Let (T,2,jz) be a finite positive measure space and X be a 
 
 separable Banach space. Let {f : A £ A} , (A is a directed set) be a net in 
 
 L (/i,X), 1 < p < co such that f converges weakly to f € L (/i,X). Suppose that 
 p A p 
 
 V 
 
 for all A G A, f>(t) G F(t) /x-a.e., where F : T -* 2 is a weakly compact, 
 integrably bounded, convex, nonempty valued correspondence. Then we can extract 
 a sequence (f : n=l , 2 , . . . } from the net {f : A G A} such that: 
 
 A A 
 
 (i) f converges weakly to f and 
 
 "A 
 n 
 
 (ii) f(t) G con w-Ls{f (t)} p-a.e. 
 
 n 
 
 As an immediate conclusion of Theorem 3 . 1 we can obtain the following 
 
 generalization of Theorem 1 in Khan-Majumdar [12]. 
 
 Corollary 3.1 : Let (T.E./ii) be a finite positive measure space and X be a 
 
 separable Banach space. Let (f : n-1 , 2 , . . . } be a sequence of functions in 
 L (fi,X), 1 < p < <» such that f converges weakly to f G L (/x,X). Suppose that 
 
y 
 
 for all n, (n-1,2,...), f (t) e F(t) /i-a.e., where F : T - 2 is a weakly 
 
 n 
 
 compact, integrably bounded, nonempty valued correspondence. Then 
 
 f(t) G con w-Ls{f (t)} /i-a.e 
 n 
 
 Corollary 3.1 generalizes Theorem 1 of Khan-Majumdar [12] in several 
 directions. In particular, the measure space (T,2,/t) need not be atomless or 
 complete, the sequence {f : n— 1,2,....) need not be in a fixed weakly compact 
 subset of X and finally the sequence (f : n=l , 2 , . . . } need not lie only in 
 
 i^Oi.x). 
 
 Using Corollary 3 . 1 we can prove the following dominated convergence result 
 for the set of integrable selections. 
 
 Theorem 3.2 : Let (T,2,/i) be a complete finite positive measure space and X 
 
 be a separable Banach space. Let <p : T -» 2 (n=l,2, . . .) be a sequence of closed 
 
 valued and lower measurable correspondences such that: 
 
 (i) For each n, (n=l,2,...), <p (t) c F(t) /i-a.e., where F : T -+ 2 is an 
 
 n 
 
 integrably bounded weakly compact, convex, nonempty valued correspondence, 
 
 (ii) <p (t) -» <p(t) /i-a.e., and 
 n 
 
 (iii) </>(•) is convex valued. 
 
 Then 
 
 n 
 
 As a Corollary of Theorem 3 . 2 we can obtain a donimated convergence result 
 for the integral of a correspondence. 
 
Corollary 3.2 : Let <p : T -* 2 (n— 1,2,...) be a sequence of closed valued 
 and lower measurable correspondences satisfying all the assumptions of Theorem 
 3.2. Then 
 
 JV n "* JV- 
 
 The above corollary may be seen as an extension of a result of Aumann 
 
 [3, Theorem 5, p. 3] to correspondences taking values in a separable Banach 
 
 i 
 space. Recall that in [3], X = R . 
 
 A version of the above dominated convergence result for the integral of a 
 correspondence, has been obtained by Pucci and Vitillaro [16]. In their paper 
 the upper and lower limits of a sequence of correspondences were defined in 
 terms of support functions. Moreover, they assumed that X is a separable 
 reflexive Banach space, and that (T,S,/i) is atomless. Hence, their result does 
 not subsume ours. Finally, Corollary 3.2 generalizes Theorem 5.2 in Yannelis 
 [22], where it was assumed that (T,S,/i) is atomless. 
 
 It should be noted that Theorem 3.2 follows easily from Lemmata 5.1 - 5.3 
 (see section 5) which are w-Ls and s-Li versions of the Fatou Lemma for the set 
 of integrable selections. In particular, Lemma 5.1, i.e., the w-Ls version of 
 the Fatou Lemma is a direct consequence of Corollary 3.1 and it can be easily 
 shown that it implies the w-Ls versions of the Fatou Lemma for the integral of a 
 function or correspondence, obtained by Khan-Majumdar [12], Balder [4] and 
 Yannelis [22] . 
 
 We can now turn to the proofs of our main theorems. 
 
4. PROOF OF THEOREM 3.1 
 
 We begin by stating the following result of Artstein which will be used for 
 the proof of Proposition 4.2 below: 
 
 Proposition 4.1 : Let (T,E,/i) be a finite positive measure space and let f : 
 T -.R*. C-1.2,...) be a uniformly Integrable sequence of functions converging" 
 
 weakly to f. Then, 
 
 f(t) G con w-Ls{f (t)} u-a.e 
 n 
 
 Proof : See [2, Proposition C, p. 280] 
 
 Proposition 4.2 : Let (T,S,/i) be a finite positive measure space and X be a 
 
 separable Banach space whose dual X* has the RNP. Let {f : n=l , 2 , . . . ) be a 
 
 sequence in L (u,X), 1 < p < °° such that f converges weakly to f G L (u,X). 
 p n p 
 
 Suppose that for all n, (n=l,2,...), f (t) G F(t) u-a.e. where F : T -> 2 is a 
 weakly compact nonempty valued correspondence. Then 
 
 f(t) G con w-Ls{f (t)} u-a.e 
 n 
 
 TV- 
 Proof: Since f converges weakly to f and X has the RNP, for any cp G 
 
 * * 1 1 
 
 (L (u,X)) - L (u,X ) (where - + - = 1) , we have that <tp,£ > = 
 p q P q n 
 
 f_< <p(t),f (t) > du(t) converges to <<p,f> = L<<p( t) , f ( t)> dAt(t). Define the 
 In 1 
 
 functions h : T - R and h: T -» R by h (t) = <<p(t), f (t)> and h(t) = <<p(t), 
 n n n 
 
 f(t)> respectively. Since for each n, f (t) G F(t) n-a.e. and F( • ) is weakly 
 
 n 
 
 compact, h is bounded and uniformly integrable. Also, it is easy to check that 
 h converges weakly to h. In fact, let g G L (/i,R) and let M = llgll^- then, 
 
|/ T g(t)(h n (t)-h(t))d/i(t)| = i; T g(t)«<p(t), f n (t)> - «p<t), f(t)>)d M (t)| 
 
 (4.1) < M|<<p,f > - <tp,f>\ 
 
 n 
 
 and (4.1) can become arbitrarily small since as it was noted above <<p,f > 
 converges to <<p,f>. 
 
 By Proposition 4.1, we have that /x-a.e., h(t) e con w-Ls{h (t) } c 
 
 con w-Ls{h (t) } , i.e., /i-a.e. , <<p(t) , f(t)> G con w-Ls{<cp(t) , f (t)>) = 
 n n 
 
 <<p(t) , con w-Ls{f (t)}> and consequently, 
 
 (4.2) / <<p(t), f(t)>d/i(t) e J* <<p(t), x(t)>d/i(t), where x(-) is a selection fro 
 
 m 
 
 con w-Ls { f ( • ) } . 
 n 
 
 It follows from (4.2) that: 
 
 (4.3) f G S P 
 
 con w-Ls { f ) 
 n 
 
 To see this, suppose by way of contradiction that f G S_ , then by the 
 
 con w-Ls { f ) 
 n 
 
 separating hyperplane theorem (see for instance [l,p.l36]), there exists V> G 
 
 (L (/i.X))* = L (m,X*), t/> * such that <i/>,f> > sup{<0,x>: x G S^ }, 
 
 con w-Ls ( f } 
 n 
 
 i.e., J <V>(t) , f (t)>d/i(t) > J* <V>(t) ,x(t)>d/x(t) , where x( • ) is a selection from 
 
 con w-Ls{f (•)). a contradiction to (4.2). Hence, (4.3) holds and we can 
 
 conclude that f(t) G con w-Ls{f (t)} /i-a.e. This completes the proof of 
 Proposition 4.2. 
 
10 
 
 Remark 4.1 : Proposition 4.2 remains true without the assumption that X 
 
 has the RNP. The proof proceeds as follows: Since f converges weakly to f we 
 
 * 
 have that <<p , f > converges to <cp,f> for all <p e (L (u,X)) . It follows from a 
 n P 
 
 standard result (see for instance Dinculeanu [8, p. 112]) that <p can be 
 
 * 
 represented by a function : T -► X such that <ip,x> is measurable for every 
 
 x e X and ||i/>|| e L (/i,R). Hence, <<p,f > = f <V>(t) , f (t)> d^(t) and <cp,f> = 
 
 f_<0(t), f(t)>d/x(t). Define the functions h : T -* R and h: T ■* R by h (t) 
 1 n n 
 
 =<V'(t) , f (t)> and h(t) = <rb(t) , f(t)> respectively. One can now proceed as in 
 n 
 
 the proof of Proposition 4.2 to complete the argument. 
 
 We are now ready to complete the proof of Theorem 3.1 
 
 Proof of Theorem 3.1 : Denote the net {f,: A 6 A) by B . Since by assumption 
 for all AeA, f (t)eF(t) /x-a.e. where F: T -» 2 is an integrably bounded, weakly 
 compact, convex valued correspondence we can conclude that for all AeA, f lies 
 
 A 
 
 in the weakly compact set SI, (recall Diestel's theorem on weak compactness, see 
 for example [20] for an exact reference). Hence, the weak closure of B, i.e., 
 w-ciB, is weakly compact. By the Eberlein-Smulian Theorem, (see [ 9 , p. 430] or 
 
 [1, p. 156]), w-ciB is weakly sequentially compact. Obviously the weak limit of 
 
 2 
 f , i.e. f, belongs to w-ciB. From Whitley's theorem [1, Lemma 10-12, p. 155], 
 
 we know that if f e w-ciB, then there exists a sequence (f : n=l , 2 , . . . ) in B 
 
 A 
 
 n 
 such that f converges weakly to f . Since the sequence (f : n=l , 2 , . . . } 
 
 A A 
 
 n n 
 
 satisfies all the assumptions of Proposition 4.2 and Remark 4 . 1 we can conclude 
 
 that f(t) G con w-Ls{f (t)} n-a.e. This completes the proof of the Theorem. 
 
 n 
 
11 
 
 > 
 
 5. PROOF OF THEOREM 3.2 
 
 For the proof of Theorem 3 . 2 we need to prove w-Ls and s-Li versions of Fatou's 
 Lemma for the set of integrable selections. 
 
 Lemma 5.1 : Let (T,E,/i) be a finite positive measure space and X be a 
 separable Banach space and let <p : T -» 2 , (n— 1,2,...) be a sequence of 
 nonempty, closed valued correspondences such that: 
 
 V 
 
 (i) For all n, (n-1,2,...), <p (t) C F(t) /i-a.e., where F: T - 2 is an 
 
 n 
 
 integrably, bounded weakly compact, convex, nonempty-valued correspondence. 
 Then, 
 
 w-Ls S C S 
 
 n con w-Lsip 
 n 
 
 Proof : Let x G w-Ls S , i.e., there exists x. G S , (k— 1,2, . . . ) such 
 
 °k 
 
 n 
 
 that x, converges weakly to x. We wish to know that x G S . Since x, 
 
 con w-Lscp 
 n 
 
 converges weakly to x and x, lies in a weakly compact set, it follows from 
 Proposition 4.2 that x(t) G con w-Ls{x, (t)} p-a.e. which implies that 
 
 x(t) G con w-Ls<p (t) /i-a.e. Since by assumption for each n, <p (•) lies in the 
 
 integrably bounded convex set F( • ) , we can conclude that x G S_ . This 
 
 con w-Lscp 
 n 
 
 completes the proof of the lemma. 
 
 With additional assumptions, than those in Lemma 5.1, we are now able to 
 obtain an exact w-Ls version of Fatou's Lemma for the set of integrable 
 selections . 
 
12 
 
 Lemma 5.2 : Let <p : T -» 2 , (tl— 1,2,...) be a sequence of correspondences 
 satisfying all the asumptions of Lemma 5.1. Moreover, assume that w-Ls<p (•) is 
 
 closed and convex valued. Then, 
 
 w-Ls S 1 C S 1 T 
 
 <p w-Ls<p 
 n n 
 
 Proof: It follows from Lemma 5.1 that: 
 
 (5.1) w-Ls S 1 c S 1 
 
 n con w-Lscp 
 n 
 
 Since w-Lsip (•) is closed and convex (hence weakly closed) we have that 
 
 w-Ls<p (•) = con w-Ls <p (•) and therefore, 
 n n 
 
 (5.2) S 1 _ = S 1 
 w-Ls<p 
 
 n con w-Lsoj 
 n 
 
 Combining now (5.1) and (5.2) we can conclude that w-Ls S C S 
 
 cp w-Lsa? 
 
 n n 
 
 This completes the proof of the lemma. 
 
 The result below is a s-Li version of Fatou's Lemma for the set of 
 integrable selections. It generalizes Proposition 4.2 in [3] to separable 
 Banach spaces . 
 
 Lemma 5.3 : Let (T,E,/j) be a complete finite measure space and let X be a 
 
 separable Banach space. If tp : T -*■ 2 , (n— 1,2,...) is a sequence of integrably 
 
 bounded correspondences having a measurable graph, i.e., G G E®j3(X), then, 
 
 n 
 
 S 1 T . C s-Li S 1 . 
 s-Li <p <p 
 
 n n 
 
13 
 
 Proof : Let x G S T . , i.e., x(t) E s-Li <s> (t) u-a.e., we must show that 
 
 s-Licp n 
 
 n 
 
 x E s-Li S . First note that x(t) E s-Lia? u-a.e. implies that there exists a 
 
 cp XI 
 
 n 
 
 sequence {x : n=l , 2 , . . . } such that s-lim x (t) = x(t) u-a.e. and x (t) E <p (t) 
 n n n n n 
 
 n-*» 
 
 u-a.e., which is equivalent to the fact that lim dist(x( t) ,<p (t)) - u-a.e. As 
 
 n 
 n-*» 
 
 in [17, p. 528, or 15a] for each n, (n=l,2,...) define the correspondence 
 
 A : T -» 2 X by A (t) = (y E <p (t) : II y - x(t) | < dist (x(t), <p (t)) + -}. 
 n nn nn 
 
 Clearly for all n, (n=l,2,...) and for all t E T, A (t) * d>. Moreover, A (•) 
 
 n n 
 
 has a measurable graph. Indeed, the function g: T x X ■* [- 00 , 00 ] defined by 
 
 g(t,y) = | y - x(t) - dist(x(t) , <p (t)) is measurable in t and continuous in y 
 
 and therefore by a standard result (see Himmelberg [10, theorem 2, p. 378]) 
 
 g(-,-) is jointly measurable with respect to the product a-algebra Z ® /3(X) . It 
 
 is easy to see that: 
 
 G A = {(t,y) E T x X : g(t,y) < ■ ) n G = g _1 ([— ,£]) n G . 
 n n n 
 
 Since <p (•) has a measurable graph and g( • , • ) is jointly measurable, we can 
 
 conclude that G belongs to S <S> /3(X) , i.e., A (•) has a measurable graph. By 
 a n 
 
 n 
 
 the Aumann measurable selection theorem (see for instance Himmelberg [10]) there 
 
 exists a measurable function f : T -► X such that f (t) E A (t) u-a.e. Since 
 
 n n n 
 
 x(t) E s-Li<p (t) u-a.e., lim dist(x(t), (p (t)) = u-a.e. which implies that 
 
 n-«*o 
 
 lim I f (t) - x(t) I = u-a.e. Since f (t) E <p (t) u-a.e. and <p (•) is 
 n-*oo » n " n n n 
 
 integrably bounded, by the Lebesgue dominated convergence theorem (see 
 
 Diestel-Uhl [7, p. 45]), f (•) is Bochner integrable , i.e., f E LAn, X). 
 
 n n 1 
 
 Hence, x E s-Li S and this completes the proof of the lemma. 
 
 We are now ready to complete the proof of Theorem 3.2 
 
14 
 
 Proof of Theorem 3.2 : First note that since for each n, (n=l,2,...) cp ( • ) 
 
 n 
 
 is closed valued and lower measurable. G E S ® ,5(X) , (see [10, Theorem 3.5]), 
 
 i.e.. tp (•) has a measurable graph and so does s-Li <p (•)• Now if 
 
 <p(t) — s-Li cp (t) — w-Ls ip (t) /i-a.e., it follows from Lemmata 5.2 and 5.3 that 
 n n 
 
 S 1 - S 1 _ . C s-LiS 1 C w-Ls S 1 C S 1 T = S . 
 s-Liv <P <P w-Lscp <p 
 
 n n n n 
 
 Therefore 
 
 S" = s-Li S = w-Ls S 
 
 n n 
 
 ar.d we can conclude that S -»• S . This completes the proof of the Theorem, 
 
 Proof of Corollary 3.2 : Define the mapping V : L..(/i,X) ■+ X by V>(x) ■ 
 J"x(t)d/x(t) . From Theorem 3 . 2 we have that: 
 
 (5.3) S = s-Li S 1 = w-Ls S 1 . 
 tp tp ip 
 
 n n 
 
 Taking into account (5.3), it follows directly from the definition of the 
 integral of a correspondence that: 
 
 tfCS 1 ) = l*(x): x G S 1 } = JV(t)d M (t) = V>(s-Li S 1 ) = s-LiJV(t)d/x(t) = 
 
 tp S tp XI 
 
 V»(w-Ls S 1 ) = w-LsJV (t)d/i(t), 
 
 % 
 
 i.e. , 
 
 as was to be shown. 
 
15 
 
 6. CONCLUDING REMARKS 
 
 Remark 6.1 : If (T,E,/i) in Lemma 5.1 is assumed to be atomless , then by 
 virtue of Result 2 in [16] one can obtain a generalized version of Fatou's Lemma 
 proved in Khan-Majumdar [12]. The proof is similar with that in [12]. 
 
 Remark 6.2: In finite dimensional spaces Balder [5] has shown that the 
 Chacon biting lemma (see [5] for a reference) can be used to generalize 
 Schmeidler's [19] version of Fatou's Lemma in several dimensions. Recently, 
 Balder [6] has extended the biting lemma to L^(n,X) where X is a reflexive 
 Banach space. It is of interest to know whether Balder 's extension of the 
 biting lemma can be used to prove Lemma 5.1, or even versions of Theorem 3.1. 
 
16 
 
 FOOTNOTES 
 
 1. Note that the set S_ is nonempty. In fact, since w-Ls{f } is 
 
 con w-Ls { f } 
 n 
 
 lower measurable and nonempty valued so is con w-Ls{f }. Hence, con w-Ls{f } 
 admits a measurable selection (recall the Kuratowski and Ryll-Nardzewski 
 measurable selection theorem) . Obviously the measurable selection is also 
 
 integrable since con w-Ls{f } lies in a weakly compact subset of X. Therefore, 
 
 we can conclude that S_ is nonempty. 
 
 con w-Ls{ f } 
 n 
 
 2. See also Kelley-Namioka [11, exercise L, p. 165 
 
17 
 
 REFERENCES 
 
 [I] Aliprantis, CD. and . Burkinshaw, Positive Operators , Academic Press, 
 New York, 1985. 
 
 [2] Artstein, Z., "A Note on Fatou's Lemma in Several Dimensions," J . 
 
 Math. Econ . 6 (1979), 277-282. 
 [3] Aumann, R.J., "Integrals of Set-Valued Functions," J. Math. Anal. Appl . 
 
 12 (1965), 1-12. 
 [4] Balder. E. J., "Fatou's Lemma in Infinite Dimensions," J. Math. Anal. Appl. 
 
 (to appear) . 
 [5] Balder, E. J., "More on Fatou's Lemma is Several Dimensions," Canadian 
 
 Mathematical Bulletin (to appear) . 
 [6] Balder, E.J. "Short Proof of an Existence Result of V.L. Levin," Bulletin 
 
 of the Polish Academy of Sciences (to appear). 
 [7] Diestel, J. and J. Uhl , Vector Measures . Math. Surveys, No. 15, 
 
 American Mathematical Society, Providence, RI 1977. 
 [8] Dinculeanu, N. , "Linear Operations on L -Spaces,": in D. Tucker and H. 
 
 Maynard (eds.), Vector and Operator Valued Measures and Applications . 
 
 Academic Press, New York, 1973. 
 [9] Dunford, N. and J.T. Schwartz, Linear Operators , Part I, Interscience , 
 
 New York, 1958. 
 [10] Himmelberg, C.J., "Measurable Relations," Fund. Math. . 87 (1975), 53-72. 
 
 [II] Kelley, J. and I. Namioka, Linear Topological Spaces , Springer, New York, 
 1963. 
 
 [12] Khan, M. Ali and M. Majumdar, "Weak Sequential Convergence in L (/j,X) and 
 an Approximate Version of Fatou's Lemma. " J. Math. Anal. Appl. 114 (1986), 
 569-573. 
 
18 
 
 [13] Khan, M. Ali and N. C. Yannelis, "Equilibria in Markets with a Continuum 
 
 of Agents and Commodities" (mimeo) , 1986. 
 [14] Khan, M. Ali, "On Extensions of the Cournot-Nash Theorem," in Advances in 
 
 Equilibrium Theory . Lecture Notes in Economics and Mathematical Systems 
 
 No. 244, CD. Aliprantis et all, (eds.) Springer-Verlag, Berlin 
 
 1985. 
 [15] Khan M. Ali, "Equilibrium Points on Nonatomic Games over a Banach Space," 
 
 Trans. Amer. Math. Soc . 29 (1986), 737-749. 
 [15a] Lucchetti, R. , N.S. Papageorgiou and F. Patrone , "Convergence and 
 
 Approximation Results for Measurable Multifunctions , " Proc . Amer. Math. 
 
 Soc. . 100, (1987), 551-556. 
 [16] Pucci P. and G. Vitillaro, "A Representation Theorem for Aumann 
 
 Integrals," J. Math. Anal. Appl . 102 (1984), 86-101. 
 [17] Ricceri, B. "Sur 1' approximation des Selections Mesurables," C. R. Acad. 
 
 Sc. Paris , t. 295, (1982), 527-530. 
 [18] Rustichini A., "A Counterexample to Fatou's Lemma in Infinite Dimensions," 
 
 Department of Mathematics, University of Minnesota, 1986. 
 [19] Schmeidler, D., "Fatou's Lemma in Several Dimensions," Proc. Amer. 
 
 Math. Soc. . 24 (1970), 300-306. 
 [20] Yannelis, N.C., "Fatou's Lemma in Infinite Dimensional Spaces," Proc , 
 
 " Amer. Math. Soc. . 102, (1988), 303-310. 
 [21] Yannelis, N.C., "Equilibria in Noncooperative Models of Competition," J . 
 
 Econ. Theory . 41 (1987), 96-111. 
 [22] Yannelis, N.C., "On the Lebesgue -Aumann Dominated Convergence Theorem in 
 
 Infinite Dimensional Spaces," Department of Economics, University of 
 
 Minnesota, 1986. 
 
HECKMAN _l 
 UNDERY INC. |§| 
 
 JUN95 
 
 md-To-Plca*? N.MANCHESTER, 
 INDIANA 46962