: " 
 
 BEBR 
 
 FACULTY WORKING 
 PAPER NO. 1496 
 
 On the Upper and Lower Semicontinuity 
 of the Aumann Integral 
 
 Nicholas C. Yannelis 
 
 it968 
 
 College of Commerce and Business Administration 
 Bureau of Economic and Business Research 
 University of Illinois. Urbana-Champaign 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 1496 
 
 College of Commerce and Business Administration 
 
 University of Illinois at Urbana- Champaign 
 
 September 1988 
 
 On the Upper and Lower Semicontinuity of the Aumann Integral 
 
 Nicholas C. Yannelis, Associate Professor 
 Department of Economics 
 
Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/onupperlowersemi1496yann 
 
ON THE UPPER AND LOWER SEMICONTINUITY OF THE AUMANN INTEGRAL 
 
 by 
 
 Nicholas C. Yannelis > 
 
 Abstract : Let (T,x»y) be a finite measure space, X be a Banach space, 
 P be a metric space and let L.(y,X) denote the space of equivalence 
 classes of X-valued Bochner integral functions on (T,x,u). We show 
 that if ij) : TxP -► 2 is a correspondence such that for each fixed p£P, 
 ((>(•, p) has a measurable graph and for each fixed teT, <j>(t,«) is either 
 upper or lower seraicontinuous then the Aumann integral of (j» , i.e., 
 J T 4>(t,p)d u (t) = {/ T x(t)d u (t) : x e S (p)}, where S (p) = (yeLjCy.X) : 
 y( t)e<j>( t ,p) y-a.e.}, is either upper or lower seraicontinuous in the 
 variable p as well. Our results extend those of Aumann (1965, 1976) 
 who has considered the above problem for X = R , and they have appli- 
 cations in general equilibrium and game theory. 
 
 Key words : Integral of a correspondence, upper semicontinuity , lower 
 semicontinuity , quasi upper semicontinuity, measurable selection, 
 Fatou's Lemma in infinite dimensions. 
 
 Department of Economics, University of Illinois, Champaign, IL 
 61820. 
 
 This is a revised version of ray paper entitled, "On the Lebesgue- 
 Aumann Dominated Convergence Theorem in Infinite Dimensional Spaces," 
 written in 1987. The present version has benefitted from the com- 
 ments, discussions and suggestions of Erik Balder, M. Ali Khan, 
 Jean-Fransois Mertens and Aldo Rustichini. Of course, I am respon- 
 sible for any remaining shortcomings. 
 
1. INTRODUCTION 
 
 Let T be a measure space, P be a metric space, X be a Banach space 
 
 X X 
 and (ji be a correspondence from TxP to 2 (where 2 denotes the set of 
 
 all nonempty subsets of X), such that for each fixed peP, <{>(•, p) has a 
 measurable graph and for each fixed teT, <j>(t,») is either upper or 
 lower semicontinuous. We would like to know whether the integral of 
 the correspondence <j> is either upper or lower semicontinuous as well. 
 It is the purpose of this paper to provide an answer to the above 
 question. Specifically, we show (Theorem 3.1) that integration pre- 
 serves upper semicontinuity (u.s.c.) and that, (Theorem 3.2) integra- 
 tion preserves lower semicontinuity (l.s.c). 
 
 It should be noted that the problem of whether integration preser- 
 ves u.s.c. or l.s.c. was first examined in a path breaking paper by 
 Auraann (1965, 1976), [see also Schmeidler (1970)]. However, Aumann 
 considered correspondences taking values in a finite dimensional 
 Euclidean space. It turns out, that the finite dimensional arguments 
 of Aumann cannot be readily adopted to cover Banach-valued correspon- 
 dences. In particular, Aumann's method of proof of the fact that 
 integration preserves u.s.c. is based heavily on the Lyapunov 
 Theorem, a result which is false in infinite dimensional spaces. 
 Nevertheless, for strong forms of u.s.c. correspondences (i.e., 
 weakly u.s.c. correspondences) a result analogous of that of Aumann 
 has been obtained in Yannelis (1988a) by means of the "approximate 
 version of the Lyapunov Theorem." However, the arguments in Yannelis 
 (1988a) cannot be adopted here to prove Theorem 3.1, since the 
 correspondences we consider are u.s.c. in a much weaker sense than 
 
that in the above paper and furthermore they are not convex or compact 
 valued. Hence, our arguments are of necessity quite different than 
 those in Yannelis (1988a). 
 
 Finally, we would like to note that as the work of Auraann (1965, 
 1976) was motivated by the problem of the existence of an equilibrium 
 in economies with a continuum of agents and finitely many commodities, 
 
 our work was also motivated by the same problem but it allows for a 
 
 2 
 continuum of commodities in addition to the continuum of agents. 
 
 The paper proceeds as follows: Section 2 contains notation and 
 
 definitions. Our main results are stated in Section 3 and their 
 
 proofs are collected in Section 4. Finally, in Section 5 we show that 
 
 integration preserves the weak closed graph property. 
 
 2. NOTATION AND DEFINITIONS 
 2 . 1 Notation 
 
 R denotes the n-fold Cartesian product of the set of real 
 numbers R. 
 
 conA denotes the closed convex hull of the set A. 
 
 A. 
 2' denotes the set of all nonempty subsets of the set A. 
 
 denotes the empty set. 
 
 / denotes the set theoretic subtraction. 
 
 dist denotes distance. 
 
 proj denotes projection. 
 
 If A C X, where X is a Banach space, dk denotes the norm closure 
 
 of A. 
 
-3- 
 
 If F , (n=l,2,...) is a sequence of nonempty subsets of a Banach 
 n 
 
 space X, we will denote by LsF and LiF the set of its (strong) limit 
 v . ' J n n 
 
 superior and (strong) limit inferior points respectively, i.e., 
 
 LsF = {xeX : x = lim x , x eF , k=l,2,...|, and 
 
 k-voo k k k 
 
 LiF = {xeX : x = lim x , x eF , n=l,2,...} 
 n l n n n > 
 
 n+oo 
 
 2.2 Definitions 
 
 Y 
 Let X and Y be sets. The graph of the correspondence <j> : X ->• 2 
 
 is denoted by G = |(x,y) e XxY : y e <j>(x)l . Let (T,t,u) be a 
 
 complete, finite measure space, and X be a separable Banach space. 
 
 The correspondence $ : T + 2 is said to have a measurable graph if 
 
 G ex® 3(X), where g(X) denotes the Borel a~algebra on X and 
 
 ♦ 
 
 X 
 denotes product a~algebra. The correspondence $ : T •*■ 2 is said to 
 
 be lower measurable if for every open subset V of X, the set 
 
 {teT : (j>(t)^V * 0| is an element of -[•• Recall [see for instance 
 
 Himraelberg (1975), p. 47 or Debreu (1966), p. 359] that if 4, : T + 2 X 
 
 has a measurable graph, then <j> is lower measurable. Furthermore, if 
 
 <}>(•) is closed valued and lower measurable then A : T * 2 has a 
 
 measurable graph. A well-known result of Aumann (1967) which will be 
 
 of fundamental importance in this paper, [see also Hiramelberg (1975), 
 
 Theorem 5.2, p. 60] says that if (T,t,u) is a complete, finite measure 
 
 space, X is a separable metric space and <j> : T •► 2 is a nonempty 
 
 valued correspondence having a measurable graph, then (j>(») admits a 
 
 measurable selection , i.e., there exists a measurable function 
 
 f : T * X such that f(t)e<j>(t) y-a.e. 
 
-4- 
 
 We now define the notion of a Bochner integrable function. We 
 will follow closely Diestel-Uhl (1977). Let (T, T , U ) be a finite 
 measure space and X be a Banach space. A function f : T + X is called 
 
 simple if there exist x. , x_ , ..., x in X and a,, a , •••> a in x 
 
 I Z n 1 Z n 
 
 n 
 
 such that f = £ x.y , where y (t) = 1 if tea. and y (t) = if 
 . . i*a. *a. i *a. 
 
 i=l 11 i 
 
 t£a.* A function f : T + X is said to be y- measurable if there exists 
 
 a sequence of simple functions f : T > X such that liraiif (t) - f(t)|| 
 
 n n 
 
 n->-<» 
 
 = for almost all teT. A ^-measurable function f : T + X is said to 
 
 be Bochner integrable if there exists a sequence of simple functions 
 
 If : n=l,2,...| such that 
 
 lim/ T llf n (t) - f(t)||d u (t) = 0, 
 
 n>oo 
 
 In this case we define for each Eex the integral to be J f(t)d|j(t) = 
 
 limLf (t)du(t). It can be shown [see Diestel-Uhl (1977), Theorem 2, 
 
 J E n M 
 n-M» 
 
 p. 45] that, if f : T ->• X is a y-raeasurable function then f is Bochner 
 integrable if and only if J II f ( t )|| dp ( t) < «. We denote by L.(y f X) the 
 space of equivalence classes of X-valued Bochner integrable functions 
 x : T ->■ X normed by 
 
 lixil = / _yx(t)ndy(t). 
 
 It is a standard result that normed by the functional n • ii above, 
 L.(y,X) becomes a Banach space, [see Diestel-Uhl (1977), p. 50]. 
 
 We denote by S the set of all X-valued Bochner integrable selec- 
 tions from q> : T + 2 , i.e. , 
 
 S - (xel^Cy.X) : x(t) e <j)(t) ji-a.e.} 
 
-5- 
 
 Moreover, as in Auraann (1965) the integral of the correspondence 
 $ : T * 2 is defined as follows: 
 
 / T <j,(t)d u (t) = {/ T x(t)d y (t) : x e S }. 
 
 In the sequel we will denote the above integral by (<j>. Recall that 
 the correspondence $ : T + 2 is said to be integrably bounded if 
 there exists a map heL .(u,R) such that sup| n xii : xe<j>(t)} _< h(t) y-a.e. 
 Moreover, note that if T is a complete measure space, X is a separable 
 Banach space and <j> : T -»• 2 is an integrably bounded, nonempty valued 
 correspondence having a measurable graph, then by the Aumann 
 measurable selection theorem we can conclude that S is nonempty and 
 therefore f T <f>( t)dy( t ) is nonempty as well. 
 
 Let A , (n=l,2,...) be a sequence of nonempty subsets of a Banach 
 space. Following Kuratowski (1966, p. 339) we say that A converges 
 
 in A (written as A > A) if and only if LiA = LsA = A. It may be 
 
 n J n n ' 
 
 useful to remind the reader that LiA and LsA are both closed sets 
 
 n n 
 
 and LiA C LsA [see Kuratowski (1966), pp. 336-338]. 
 n n 
 
 Let X be a metric space and Y be a Banach space. The correspon- 
 
 Y 
 dence ^ : X > 2 is said to be u.s.c. at x_eX, if for any neighborhood 
 
 N((()(x_)) of <}>(x n ), there exists a neighborhood N(x n ) of x n such that 
 
 for all xeN(x n ), <j>(x) C N(<f>(x n )). We say that <j> is u.s.c. if $ is 
 
 u.s.c. at every point xeX. Recall that this definition is equivalent 
 
 to the fact that the set IxeX : <{>(x) C V| is open in X for every open 
 
 subset V of Y, [see for instance Kuratowski (1966), Theorem 3, p. 
 
 176]. 
 
-6- 
 
 Let v be a small positive number and let B be the open unit ball 
 
 Y 
 in Y. The correspondence <f> : X -»■ 2 is said to be quasi upper- 
 
 semicontinuous (q.u.s.c.) at xeX, if whenever the sequence x^, 
 
 n' 
 (n=l,2,...) in X converges to x, then for some n~ , <j>(x ) C <j>(x) + vB 
 
 for all n _> n„ . We say that <j> is q.u.s.c. if <p is q.u.s.c. at every 
 point xgX. It can be easily checked that if <j> is compact valued, 
 quasi upper-seraicontinuity implies upper-semicontintuity and vice- 
 versa. 
 
 Let now P and X be metric spaces. The correspondence F : P * 2 
 is said to be l.s.c. if the sequence p , (n=l,2,...) in P converges to 
 peP, then F(p) C LiF(p ). Finally recall that the correspondence 
 F : P * 2 is said to be continuous if and only if it is u.s.c. and 
 l.s.c. 
 
 With all these preliminaries out of the way we can now turn to the 
 statements of the main theorems. 
 
 3. THE MAIN THEOREMS 
 We now state our main results: 
 
 Theorem 3.1 : Let (T,t»y) be a complete, finite measure space, P 
 
 X 
 be a metric space and X be a separable Banach space. Let \\> : TxP -► 2 
 
 be a nonempty valued, integrably bounded correspondence, such that for 
 
 each fixed teT, ij;(t,0 is q.u.s.c. and for each fixed peP, ^(«,p) has 
 
 a measurable graph. Then 
 
 J\\>( t , • ) is q.u.s.c. 
 
•7- 
 
 Theorem 3.2 : Let (T,x,y) be a complete, finite measure space, X 
 be a separable Banach space and P be a metric space. Let <{> : TxP ->■ 2 
 be an integrably bounded correspondence such that for each fixed teT, 
 4>(t,») is l.s.c. and for each fixed peP, <j>(«,p) has a measurable 
 graph. Then 
 
 f<J>( t , • ) is l.s.c. 
 
 Remark 3.1 : If in addition to the assumptions of Theorem 3.1, it 
 is assumed that fi|»(t,«) is compact valued, then we can conclude that 
 J\l>( t , • ) is u.s . c. 
 
 Remark 3.2 : If in Theorem 3.1 we add the assumption that ^(«,0 
 is convex valued and that for all (t,p)eTxP, ^(t,p) C K, where K is a 
 weakly compact, convex, nonempty subset of X, then it follows from 
 Lemma 4.1 (see next section) that J T ^( t , • )dy ( t ) is weakly compact 
 valued and we can conclude that f T ^( t , • )dy ( t ) is weakly u.s.c, i.e., 
 the set (peP : j T \\>( t ,p)d(j( t ) C V| is open in P for every weakly open 
 subset V of X. Hence, from Theorem 3.1 we can obtain a version of 
 Theorem 4.1 in Yannelis (1988a) which does not require (T,t,jj) to be 
 atomless. 
 
 The Corollaries below follow directly from Theorems 3.1, 3.2 and 
 Remark 3.1. They extend some results of Aumann (1965, Theorem 5, and 
 Corollary 5.2) to separable Banach spaces. 
 
 Corollary 3.1 : Let (T,t,u) be a complete, finite measure space, P 
 be a metric space and X be a separable Banach space. Let ^ : TxP -► 2 
 be an integrably bounded, nonempty valued correspondence such that for 
 
-8- 
 
 each fixed peP> ^(*,p) has a measurable graph and for each fixed teT, 
 t}»(t,0 is continuous. Moreover, suppose that f-tyi t , . )dy ( t ) is compact 
 valued. Then 
 
 f T i|>( t , • )dy ( t ) is continuous. 
 
 3 
 
 Corollary 3.2 : Let (T,x,u) be a complete, finite measure space 
 
 and X be a separable Banach space. Let $ : T -»■ 2' , (n=l,2,...) be a 
 sequence of integrably bounded, nonempty valued correspondence having 
 a measurable graph, such that: 
 
 (i) For all n, (n=l,2,...), <£ (t) C K y-a.e., where K is a com- 
 pact, nonempty subset of X, and 
 (ii) 4> n (t) + <fr(t) u - a«e. 
 Then 
 
 J T 4, n (t)d u (t) + a/ T <j,Ct)d u (t). 
 
 Moreover, if <()(•) is convex valued then 
 
 / T <j> n (t)du(t) + / T <J>(t)dy(t). 
 
 4. PROOF OF THE MAIN THEOREMS 
 4 . 1 Lemmata 
 
 For the proof of our main results we will need some preparatory 
 Lemmata. 
 
 Lemma 4.1 : Let (T,x,u) be a finite measure space and X be a 
 Banach space. Let <|> : T -► 2 be a correspondence satisfying the 
 following condition: 
 
 (i) <j)(t) C K y-a.e., where K is a compact, nonempty subset of X. 
 
-9- 
 
 Then 
 
 c&J_con$(t)dy(t) = / con<j>(t)dy ( t) 
 
 Proof : Let K = conK. Note that K is compact, [see Dunford- 
 Schwartz (1958), Theorem 6, p. 416] nonempty and convex. Hence, from 
 Diestel's theorem [Diestel (1977), Theorem 2] we have that S~ is 
 
 IS. 
 
 weakly compact in L.(y,X). Since con<j>(«) is norm closed and convex 
 
 valued so is S . It is a consequence of the Separation Theorem 
 
 con<}> ^ r 
 
 that the weak and norm topologies coincide on closed convex sets. 
 
 Hence, S is weakly closed. Since S C S~ and the latter set is 
 
 con<j) con<j> K 
 
 weakly compact we can conclude that S is weakly compact. Define 
 
 con^j 
 
 the mapping y : L (y,X) + X by y(x) = J T x(t)dy(t). Certainly y is 
 linear and norm continuous. It follows from Theorem 15 in Dunford- 
 Schwartz (1958, p. 422) that y is also weakly continuous. Therefore, 
 
 y(S ) = |y(x) : xe S \ = f_cond>(t)du(t) is weakly compact, and 
 
 1 con* l con* J J T Y 
 
 w 
 
 e can conclude that clj con$( t )dy ( t ) = j con<j>(t). This completes the 
 proof of the Lemma. 
 
 Notice that the above proof of the Lemma showed that 
 J_con<j)( t)dy ( t ) is weakly compact. Hence, the above Lemma may be seen 
 as the infinite dimensional extension of Theorem 4 of Aumann (1965). 
 
 The result below is an infinite dimensional Li version of the 
 Fatou Lemma for the set of all integrable selections from a correspon- 
 dence. 
 
 Lemma 4.2 : Let (T,t,u) be a complete, finite measure space and X 
 
 X 
 be a separable Banach space. Let $ : T -► 2 , (n=l,2,...) be a 
 
-10- 
 
 sequence of integrably bounded correspondences having a measurable 
 
 graph, i.e., G ex ® 8(X). Then 
 *n 
 
 S T . C LiS . 
 Y n Y n 
 
 Proof: See Yannelis (1988b, Lemma 5.3). 
 
 4.2 Proof of Theorem 3.1 
 
 Without loss of generality we may assume throughout the argument 
 that j dy(t) = 1. Let B be the open unit ball in X, and v be a small 
 positive number. We must show that if jp : n=l,2,...l is a sequence 
 in P converging to peP, then for a suitable n n , 
 
 / \j>(t,p )d u (t) C J rj,(t,p)dy(t) + vB for all n > n Q . 
 
 Define the mapping S : P + 2 L l (u ' X) by S (p) = jx e L,(u,X) : 
 x( t )ei|>( t ,p) y-a.e.j. Let B and B be the open unit balls in X and 
 L.(u,X) respectively. We first show that for a suitable n n , S (p ) C 
 S (p) + vB for all n >^ n~. 
 
 We begin by finding the suitable n_. Since for each fixed teT, 
 ij>(t,») is q.u.s.c. we can find a minimal M such that 
 
 (4.1) ij>(t,p ) C i|»(t,p) + 6B for all n >_ M , 
 
 where 5 = jjfa. 
 
 We now show that M is a measurable function of t. However, first we 
 
 make a few observations. By assumption for each fixed p and n, 
 
 G ,r u t0 e t ® B(X) and so does (G , n, cd ) C » (where S C denotes 
 
 i|K*,p ) + 6o ijn«,p ; + 5B 
 
-li- 
 
 the complement of the set S). It is easy to see that 
 G t sH (G,/ ^ r J C e t ® 6(X). Therefore, the set 
 
 V« ,P) *(• >P n ) + 5B 
 
 n 
 
 U = {(t,x) £ TxX : (t,x)e G , .O (G , w*r )C } 
 
 belongs to t® S(X). 
 
 It follows from the projection theorem [see for instance Hildenbrand 
 
 (1974), p. 44] that 
 
 proj T (U)e T . 
 
 Notice that, 
 
 proJ T (U) = {t e T : ij>(t,p) < i|>(t,p n ) + 6B} 
 
 - {t e T : ij;(t,p)/( l j,(t,p ) + 6B) * 0} . 
 
 By virtue of the raeasurability of the above set we can now conclude 
 that M is a measurable function of t. In particular, simply notice 
 that, 
 
 {t e T: M t =m( =r\{t e T: i|»(t,p n ) C <{;( t , p)+6B} C\ { teT: ^(t.p^) < <j,( t , p)+6B] 
 n>m 
 
 We are now in a position to choose the desired n~. Since ^(»,») is 
 
 integrably bounded there exists heL (p,R) such that for almost all 
 
 teT, sup{ II xii : x £ i|)(t,p)} _< h(t) for each p e P. 
 
 Choose 6 such that if U (S) < 5 , (S C T) , then J h(t)dy(t) < y. 
 
 Since M is a measurable function of t, we can choose n n such that 
 
 y({teT : M _> n~} ) < 6 . . This is the desired n~ . Let n _> n n 
 
 yeSCp ). We must show that yeS (p) + vB. 
 ^ n ty 
 
 and 
 
-12- 
 
 By assumption, for each fixed peP, ^(»,p) has a measurable graph 
 and i^(«,«) is nonempty valued. Hence, by the Aumann measurable selec- 
 tion theorem there exists a measurable function f. : T -»• X such that 
 
 X 
 f,(t) e t|)(t,p) y-a.e. Define the correspondence 8 •' T > 2 by 
 
 9(t) = ({y(t)( + 5B) O i|»(t,p). It follows from (4.1) that for all 
 
 teT n = {t : M _< n-j , e(t) * 0. Moreover, 9(») has a measurable 
 
 graph. Another application of the Aumann measurable selection theorem 
 
 allows us to guarantee the existence of a measurable function 
 
 f 9 : T + X such that f„(t) e 6(t) u -a.e. Define f : T + X by 
 
 f (t) for t^T 
 f 2 (t) for teT Q . 
 
 Then f(t)e^(t,p) y-a.e. and since ij;(»,0 is integrably bounded we can 
 conclude that feS (p). If we show that ii f-yii < v then ygS (p) + vB 
 and we will be done. But this is easy to see. We have 
 
 Hf-yil = St/t "Mt) ~ y(t)lldy(t) + f_ llf ? (t)-y(t)||dy(t) 
 i/i i Q 
 
 < 2/ , h(t)d u (t) + / 6d u (t) 
 i/l 
 
 This completes the proof of the fact that, if the sequence 
 {p : n=l,2,...| in P converges to peP, then for a suitable n_ 
 
 (4.2) S (p ) C S (p) + V B for all n > n n . 
 
 \\> n ijj — U 
 
 Define now the mapping y : L.(y,X) + X by y(x) = J x(t)dy(t). It 
 follows from (4.2) that for all n _> n~ , 
 
-13- 
 Y(8 t (P B )) - (Y(x) : xeS^(p n )} 
 
 = f-l|>(t,p )dy(t) C Y (S(p) + V B) = y(S,(p)) + Y (vB) 
 
 = / T ^(t > p)d M (t) + V B 
 Hence, 
 
 Li|»(t,p n )dy(t) C J T 4,(t,p)d u (t:) + vB for all n _> n Q . 
 i.e., f—iji(t, • )dy(t) is q.u.s.c. as was to be shown. 
 4.3 Proof of Theorem 3.2 
 
 T ( X ) 
 We first show that the correspondence S : P -»- 2 1 ^ ' " defined by 
 
 S x (p) ■ (yeL ( U ,X) : y(t)ed»(t,p) p-a.e.l 
 d> i 
 
 is l.s.c. 
 
 To see this, let (p : n-1,2,...} be a sequence in P converging to 
 
 peP. We must show that S (p) C LiS (p ). Since by assumption for 
 
 d> (t> n 
 
 each fixed teT, <f>(t,») is l.s.c. we have that d>(t,p) CLi<j>(t,p ) for 
 all teT, and therefore, 
 
 (4 ' 3) yP )CS Ll/Pn>- 
 
 It follows now from Lemma 4.2 that (4.3) can be written as 
 
 S x ( P } C S r^(P ) C LiS (p ) 
 <f> Lid) n $ r n 
 
 Hence , 
 
 S (O is l.s.c. 
 
-14- 
 
 Define now the mapping y : L.(y,X) + X by y(x) = | x(t)dy(t). 
 Then y i- s linear and norm continuous. Notice that y(S (p)) = 
 fy(x) : xeS (p)} = f T <J>(t ,p)dy ( t). Since S (.) is l.s.c. so is y(S ), 
 
 » ffl 1 <P d> 
 
 i.e., f rj>( t , • )dp( t ) is l.s.c. as was to be shown. This completes the 
 proof of Theorem 3.2. 
 
 4 .4 Proof of Corollary 3.2 
 
 We begin by proving an approximate version of the Fatou Lemma in 
 infinite dimensions [see also Balder (1987), Khan-Majuradar (1986) and 
 Yannelis (1988a) for w-Ls versions of this Lemma], which may be con- 
 sidered as an extension of the finite dimensional Fatou-type Lemmata 
 obtained in Auraann (1965), Artstein (1979), Balder (1984), 
 Hildenbrand-Mertens (1971), Rustichini-Yannelis (1986), and Schmeidler 
 (1970). 
 
 Lemma 4.3 : Let (T,x,u) be a complete, finite measure space and X 
 
 X 
 be a separable Banach space. Let <jj : T -> 2 , (n=l,2,...) be a 
 
 sequence of nonempty valued, graph measurable and integrably bounded 
 
 correspondences, taking values in a compact, nonempty subset of X. 
 
 Then 
 
 LsJ $ (t)du(t) C c2,/ T Ls<j> n (t)d u (t). 
 
 Moreover, if Ls<+> (•) is convex valued, then 
 Y n 
 
 Ls/ T <fr (t)dy(t) C /Ls<j> (t)dy(t). 
 
 Proof : Denote by P the interval [0,1). Define the correspondence 
 ijj : TxP * 2 by 
 
-15- 
 
 . <b (t) 
 4,(t, P ) = V <}> n (t) U * n+1 U) 
 
 if —-7- < D < ~ 
 
 n+1 r n 
 
 if p = 
 
 n+1 
 
 I Ls<j> (t) 
 Y n 
 
 if p = 0. 
 
 It can be easily checked that for each fixed teT, *(t,«) is u.s.c. and 
 
 that for each fixed peP, <J>(*>p) has a measurable graph. Moreover, * 
 
 is integrably bounded. Hence, * satisfies all the assumptions of 
 
 Theorem 3.1 and thus, j tyi t , • )dy ( t) is q. u.s.c. Let now 
 
 x e Ls („<$) (t)du(t), i.e., there exists x such that lim x = x, 
 J T r n M n. , n. 
 
 k k-*» k 
 
 x £ f_* (t)dy(t), (k=l,2,...). We wish to show that 
 
 n k \ 
 
 x e ci/ T Ls4> n (t)d u (t). 
 
 Since j ty( t , • )dy( t) is q. u.s.c. it follows that if p converges 
 
 n k 
 
 to then J ^(t,p )du(t) C J \\>( t ,0)d u ( t) + vB for all sufficiently 
 
 k 
 
 large k. Consequently, x e f T ^( t ,0)dy ( t) + v>B for all sufficiently 
 
 n k 
 
 large k and therefore, xecJ, f_il(( t ,0)dy( t) = cjtf_LsA (t)dn(t) as was to 
 
 J T J T n 
 
 be shown. If now Ls* (•), is convex valued (recall that Ls* (•) is 
 
 T n Y n 
 
 closed valued as well) it follows from Lemma 4.1 and the first conclu- 
 sion of Lemma 4.3 that 
 
 Ls/ T $ dy(t) C c£j_LsA (t)dy(t) = J Ls^ (t)dy(t). 
 
 The proof of the Lemma is now complete. 
 
 We are now ready to complete the proof of Corollary 3.2. Notice 
 first that it follows from Lemma 4.2 that 
 
 (4.4) jLi* C Li J a. . 
 
-16- 
 
 To see this define the linear mapping y : L.(jj,X) * X by y(x) = 
 
 f_x(t)d u (t). Note that yiS.. ) = jv(x) : x e S T . 1 = [Li* and hence 
 > T ' Li* l ' Li(b J J y n 
 
 *n T n 
 
 by virtue of Lemma 4.2 we can conclude that y(S. . ) C y(LiS ) = 
 
 Lid) A 
 
 Y n T n 
 
 {•y(x) : xeLiS } = Lifd> . This completes the proof of (4.4). Since 
 
 L d) J n 
 
 Y n 
 
 by assumption <j> (t) -»- d>(t) y-a.e., i.e., <j> ( t ) = Lid) (t) = 
 
 Lsd) (t) y-a.e., it follows from Lemma 4.3 and the expression (4.4) 
 
 above that: 
 
 (4.5) /d> = /Li<fr n C Li/«|, n C Ls/d, n C cfc/Ls^ = c£/<J,. 
 
 Therefore , 
 
 c£/ T d>(t)d u (t) = LiJ <|> (t)du(t) = LsJ $ (t)d u (t), 
 i.e. , 
 
 / T <)> (t)d u (t) + c£/ T cf,(t)d u (t). 
 
 If now <j>(«) is convex valued, (4.5) can be written (recall the second 
 conclusion of Lemma 4.3) as: 
 
 /♦ = / Li r n C Li /<l> n C Ls /<l> n C / Ls <J> n = /♦• 
 
 Thus, 
 
 / T <J>(t)dy(t) = Li/ ^ (t)d u (t) = LsJ <j» (t)du(t), 
 
 i.e., 
 
 / T <f> n (t)d u (t) * / T 4,(t)d u (t), 
 and this completes the proof of Corollary 3.2 
 
-17- 
 
 5. ON THE WEAK CLOSED GRAPH PROPERTY OF THE AUMANN INTEGRAL 
 
 Let {A : n=l,2,...} be a sequence of nonempty subsets of a Banach 
 space X, and denote by w-LsA the set of its weak limit superior 
 points , i.e. , 
 
 w-LsA = {xeX : x = w-limx , x eA , k=l,2,...l. 
 n i . n, n . n. J 
 
 k->-ao k k k 
 
 Let (T,f,p) be a complete finite measure space, P be a metric 
 space and X be a separable Banach space. The correspondence 
 ijj : TxP + 2 is said to have a weakly closed graph if 
 
 w-Ls^(t,p ) C ij»(t t p) y-a.e., whenever the sequence |p : n=l,2,...| in 
 P converges to peP. 
 
 The following result in Yannelis (1988b) will be used to prove 
 that if for each fixed tgT, \\>(t y *) has a weakly closed graph then so 
 does the integral of <|j(t,*)« 
 
 Lemma 5.1 : Let (T,x,y) be a finite measure space and X be a 
 separable Banach space. Let |f : n=l,2,...| be a sequence of func- 
 tions in L (ji,X), 1 _< p < oo such that f converges weakly to 
 feL (y,X). Suppose that for all n, (n=l,2,...), f (t)eF(t) y-a.e., 
 where F : T + 2 is a weakly compact, integrably bounded, nonempty 
 valued correspondence. Then 
 
 f (t)econw-Lsj f (t)} y-a.e 
 
 n 
 
 Proof : See Yannelis (1988b, Corollary 3.1). 
 
 We are now ready to state the main result of this section, which 
 generalizes Theorem 4.1 in Yannelis (1988a). 
 
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 Theorem 5.1 : Let (T,t,u) be a complete, finite measure space, P 
 be a metric space and X be a separable Banach space. Let ijj : TxP ■* 2 
 be a nonempty, closed, convex valued correspondence such that: 
 (i) for each fixed tgT, ij>(t,») has a weakly closed graph, 
 (ii) for all (t.p) E TxP, ijj(t,p) CK(t) where K : T * 2 X is an 
 integrably bounded, weakly compact and nonempty valued 
 correspondence. 
 Then 
 
 Ji|)(t,.) has a weakly closed graph. 
 
 Proof : We first show that the set-valued function 
 S : P + 2 Ll(u,X) defined by 
 
 S (p) = {xeLjCy.X) : x( t ) e ,j,( t ,p) u -a.e.}, 
 
 has a weakly closed graph, i.e., if jp : n=l,2,...| is a sequence in 
 P converging to peP» then 
 
 (5.1) w-LsS (p ) C S (p)_ 
 
 lj; n ip 
 
 To this end let xew-LsS (p ), i.e., there exists x^ , (k=l,2,...) in 
 
 \j; n K 
 
 L (u,X) such that x^ converges weakly to xeL (y,X), and x (t)e^(t,p ) 
 
 k 
 u-a.e. , we must show that xeS (p). It follows from Lemma 5.1 that 
 
 x( t )econw-Ls| x, ( t ) | u-a.e. and therefore 
 (5.2) x(t)econw-Ls^( t ,p ) u-a.e. 
 
 n 
 
 Since for each fixed teT, ip(t,») has a weakly closed graph we have 
 that: 
 
-19- 
 
 (5.3) w-Lsij>(t,p ) C ^ ( t , p) p-a.e 
 
 n 
 
 Combining (5.2) and (5.3) and taking into account the fact that \|> is 
 
 convex valued we have that x(t)eij/(t ,p) y-a.e. Since ty is integrably 
 
 bounded, we can conclude that xgS (p). This completes the proof of 
 
 the fact that S (•) has a weakly closed graph. Define the linear 
 •J* 
 
 mapping 
 
 tt : L^u.X) + X by tt(x) = /x(t)dy(t). 
 
 It follows from (5.1) that if the sequence {p : n=l,2,...j in P con- 
 verges to peP , then 
 
 Tr(w-LsS,(p )) = W(x) : x e w-LsS,(p )} 
 
 = w-LsU(t,p ) Ctt(S (p)) = Itt(x) : x e S (p)} 
 
 i.e., fij>(t:,0 has a weakly closed graph as was to be shown. 
 
 Remark 5.1 ; It can be easily shown by means of the failure of the 
 Lyapunov theorem in infinite dimensional spaces that Theorem 5.1 is 
 
 false without the convex valueness of the correspondence 
 
 X 
 \\i : TxP > 2 [see Rustichini (1987) for a complete argument]. 
 
-20- 
 
 FOOTNOTES 
 
 In general equilibrium theory, T denotes the measure space of 
 agents, X denotes the commodity space, P denotes the price space, 
 4>(t,p) denotes the demand set of agent t at prices p and the integral 
 of (j> denotes the aggregate demand set [see for instance Aumann (1966) 
 or Schmeidler (1969) or Hildenbrand (1974) who have considered the 
 
 I 
 above problem for X = R , in order to prove the existence of an 
 
 equilibrium for an economy with an atomless measure space of agents 
 
 and with finitely many commodities]. 
 
 2 
 See for instance Khan-Yannelis (1987) for the usefulness of our 
 
 results in general equilibrium theory. Also, applications of our 
 
 results in game theory are given in Balder-Yannelis (1988). 
 
 3 
 Compare with Corollary 3.2 in Yannelis (1988b) where a different 
 
 notion of convergence of sequences of set-valued functions was used. 
 
-21- 
 
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