Weak sequential convergence in Lp (, X)

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ST

BEB

FACULTY WORKING
PAPER NO. 1489

Weak Sequential Convergence in Lp (/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, fx(t) 6 F(t) u-a.e. , where F : T - 2X
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 y1 ,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 . 1yiAa. Aa. l Aa.

1=1 11 l

0 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 = 0 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"tI! fn(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 JEn

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)||pd/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.,

SP = {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 :

JT<p(t)d/i(t) = {/Tx(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

JVn "* 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,

|/Tg(t)(hn(t)-h(t))d/i(t)| = i;Tg(t)«<p(t), fn(t)> - «p<t), f(t)>)dM(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 SP

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/> * 0 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 0 : 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 S1 C S1 T

<p w-Ls<p
n n

Proof: It follows from Lemma 5.1 that:

(5.1) w-Ls S1 c S1

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) S1 _ = S1
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

S1 T . C s-Li S1 .
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)) - 0 u-a.e. As

n
n-*»

in [17, p. 528, or 15a] for each n, (n=l,2,...) define the correspondence

A : T -» 2X 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:

GA = {(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)) = 0 u-a.e. which implies that

n-«*o

lim I f (t) - x(t) I = 0 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

S1 - S1 _ . C s-LiS1 C w-Ls S1 C S1 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 S1 = w-Ls S1 .
tp tp ip

n n

Taking into account (5.3), it follows directly from the definition of the
integral of a correspondence that:

tfCS1) = l*(x): x G S1} = JV(t)dM(t) = V>(s-Li S1) = s-LiJV(t)d/x(t) =

tp S tp XI

V»(w-Ls S1 ) = 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 0 . Burkinshaw, Positive Operators, Academic Press,
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[2] Artstein, Z., "A Note on Fatou's Lemma in Several Dimensions," J .

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[12] Khan, M. Ali and M. Majumdar, "Weak Sequential Convergence in L (/j,X) and
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18

[13] Khan, M. Ali and N. C. Yannelis, "Equilibria in Markets with a Continuum

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