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Fatou's Lemma in Reflexive Banach Spaces
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
199, 748]753 Ž1996.
0172
Fatou’s Lemma in Reflexive Banach Spaces S. I. Suslov Russian Academy of Sciences, Siberian Branch, Institute of Mathematics, No¨ osibirsk, 630090, Russia Submitted by Richard M. Aron Received September 7, 1994
A version of an approximate Fatou Lemma for a uniformly integrable sequence of functions with values in a reflexive Banach space is proved. The usual assumption that this sequence is pointwisely dominated in norm by a real valued integrable function is omitted. Q 1996 Academic Press, Inc.
Fatou’s lemma in n-dimensions proved to be very useful in mathematical economics Žsee, e.g., w4x.. Recently, the work in economics with infinite dimensional spaces of commodities attracted attention to this lemma in infinite dimensions w2, 5, 6, 9x. In the present paper, we prove a version of such a lemma, an approximate undominated one in a reflexive Banach space, thus generalizing the result of w2x. Let ŽT, A, m . be the interval w0, 1x with Lebesque s-algebra A and Lebesgue measure m , X be a reflexive Banach space with the dual space X *, and f n4 be a uniformly integrable sequence from the Banach space L1ŽT, X . of Bochner integrable functions f : T ¬ X. For a sequence A n4 of non-empty subsets of X write: w y Ls A n [ x g X : x s weak lim x k , x k g A n k , k g N 4 . Here and in what follows we use the symbol w to indicate that the corresponding topological notion should be understood with respect to weak topology. Our main result is: THEOREM. The following inclusion holds w y Ls
HT f
n
: cl 748
0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
HT w y Ls f . n
FATOU’S LEMMA IN BANACH SPACES
749
Here cl stands for the norm closure and the integral of a multivalued function t ¬ w y Ls f nŽ t . is understood in the sense of Aumann, i.e., as the set
½H
T
h: H g L1 Ž T , X .
and
h Ž t . g w y Ls f n Ž t . a.e. on T
5
of integrals of all integrable selectors of this multivalued function. Since all f n are m-essentially separably valued, we can and shall assume throughout the proof that X is separable. We use the following standard notations: co represents the convex hull; B Ž r . [ x g X : < x < F r 4 for r ) 0; sŽ x*, A. [ sup² x*, x :: x g A4 for x* g X *, A ; X; gr F [ Ž t, x . : x g F Ž t .4 for multifunction F : T ¬ 2 X . We shall prove the theorem by a sequence of lemmas. For t g T write: L Ž t . [ w y Ls f n Ž t . ,
F Ž t . [ cl co L Ž t . ,
LEMMA 1. The multifunction F is measurable Ž see w8x for definition.. Proof. Let Fm n Ž t . [ w y cl Ž B Ž n . l f k Ž t . 4 kGm . ,
t g T.
Being weakly compact, every ball B Ž n. is Polish in its weak topology w3, Theorem V, 6.3x. Consequently, we can use the results of w8x and easily obtain that the sets gr Fm n , m, n g N are A = BŽ X .-measureable, where BŽ X . stands for the Borel s-algebra of subsets of X. Then, since weakly convergent sequences are bounded in X, we have: gr L s
D F gr Fm n .
ngN mgN
So gr L is a A = BŽ X .-measurable set, and hence the multifunction L has a Castaing representation w8x, easily yielding the result. For t g T, n g N, set rn Ž t . [ < f n Ž t . < q lim inf < f k Ž t . < q 1, Fn Ž t . [ B Ž rn Ž t . . l F Ž t . ,
t g T , n g N.
LEMMA 2. The sequence rn4 is uniformly integrable and FnŽ t . / 0u a.e. on T for e¨ ery n g N.
750
S. I. SUSLOV
Proof. This follows immediately since, by the scalar Fatou lemma, the function t ¬ lim inf < f k Ž t .< is integrable. For x* g X *, write
a n Ž x*, t . [ max 0, ² x*, f n Ž t . : y s Ž x*, Fn Ž t . . 4 . Note that, by Lemmas 1 and 2, the functions a nŽ x*, ?. are well-defined and measurable. LEMMA 3. The sequence a nŽ x*, ?. is con¨ ergent to zero in measure. Proof. Since the sequences rn4 and f n4 are bounded in L1ŽT, R. and L1ŽT, X . respectively, a standard by-contradiction argument yields the existence of a point t 0 g T, a positive number « , and a subsequence k n such that
a k nŽ x*, t 0 . G «
; n g N,
Ž 1.
sup < r k nŽ t 0 . < R - `,
Ž 2.
sup < f k nŽ t 0 . < X - `.
Ž 3.
ngN
ngN
Due to Ž2. and Ž3. we can assume, extracting a further subsequence if necessary, that r k nŽ t 0 . ª r , w
f k nŽ t 0 . ª x 0
Ž 4. Ž 5.
hold for some r g R and x 0 g X. Observe now, this being the key point of the proof, that Ž4. implies s Ž x*, Fk nŽ t 0 . . ª s Ž x*, B Ž r . l F Ž t 0 . . , which together with a trivial consequence of Ž5. ² x*, f k Ž t 0 . : ª ² x*, x 0 : n yields lim a k nŽ x*, t 0 . s max 0, ² x*, x 0 : y s Ž x*, B Ž r . l F Ž t 0 . 4 .
Ž 6.
The right-hand side of Ž6., however, is equal to zero because x 0 , being certainly in B Ž r ., belongs to F Ž t 0 . by definition. This contradiction with Ž1. completes the proof of the lemma.
FATOU’S LEMMA IN BANACH SPACES
751
LEMMA 4. If f n4 or a subsequence thereof con¨ erges weakly to f g L1ŽT, X ., then f Ž t . g cl co w y Ls f nŽ t .4 a.e. on T. Consequently, for e¨ ery a g w y Ls HT f n there exists a function f g L1ŽT, X . such that f Ž t . g F Ž t . a.e. on T and HT f s a. Proof. Clearly it suffices to prove the lemma when the sequence f n4 w itself converges weakly to f. So let f n ª f. By the previous lemma, the sequence a nŽ x*, ?. converges to zero in measure. Hence, by w3, Corollary III.6.13x, we can assume, extracting a subsequence if necessary, that
a n Ž x*, t . ª 0
a.e. on T ; x* g X *.
Ž 7.
For « G 0 and x* g X *, set H Ž x*, « . [ x g X :² x*, x : F « 4 . Then, by the definition of a nŽ x*, t ., we have f n Ž t . g H Ž x*, s Ž x*, Fn Ž t . . q a n Ž x*, t . .
;t g T
Ž 8.
Note now that rn4 , being uniformly integrable, is weakly relatively compact in L1ŽT, R., so that we can suppose, extracting a subsequence if w necessary, that rn ª r for some r g L1ŽT, R.. Consequently, by Mazur’s theorem, for each n g N there exist nonn l n s 1, such that negative numbers l ni , i s 1, . . . , k n , with Ý kis1 i rnX [
kn
L1
l ni rnqi ª r ,
Ý is1
f nX [
kn
L1
l ni f nqi ª f .
Ý is1
Moreover, we can suppose, extracting a subsequence if necessary, that rnX Ž t . ª r Ž t .
a.e. on T
Ž 9.
f nX Ž t . ª f Ž t .
a.e. on T
Ž 10 .
Then Ž8. yields f nX Ž t . g H Ž x*, s Ž x*, F Ž t . l B Ž rnX Ž t . . . q a nX Ž x*, t . . where
a nX
kn
Ž x*, t . [
Ý is1
l ni a nqi Ž x*, t . .
a.e. on T Ž 11 .
752
S. I. SUSLOV
But
a nX Ž x*, t . ª 0
a.e. on T
and s Ž x*, F Ž t . l B Ž rnX Ž t . . . ª s Ž x*, F Ž t . l B Ž r Ž t . . . by Ž7. and Ž9. respectively. So Ž10. and Ž11. imply f Ž t . g H Ž x*, s Ž x*, F Ž t . l B Ž r Ž t . . .
a.e. on T ; x* g X *
and consequently f Ž t. g
F H Ž xUn , s Ž xUn , F Ž t . l B Ž r Ž t . . .
a.e. on T
Ž 12 .
ngN
holds for some denumerable set xUn 4 ; X * separating the points of X. Note that the right-hand side of Ž12. is equal to F Ž t . l B Ž r Ž t .. because the latter set is weakly compact in X. Thus, f Ž t . g F Ž t . a.e. on T which, together with the evident relation HT f s w y lim HT f n s a, yields the result. Now, the next lemma completes the proof of our theorem. LEMMA 5.
It holds cl
H L s cl HT F. T
Proof. Observe that in proving Lemma 1 the A = BŽ X .-measurability of the set gr L was established. In addition, the sequence f n4 is uniformly integrable by our assumption. Then, arguing as in w1, Theorem Ax, we obtain an integrable selector of L and apply w7, Corollary 3.3x.
REFERENCES 1. Z. Artstein, A note on Fatou’s lemma in several dimensions, J. Math. Econ. 6 Ž1979., 277]282. 2. E. Balder, Fatou’s lemma in infinite dimensions, J. Math. Anal. Appl. 136 Ž1988., 450]465. 3. N. Dunford and J. T. Schwartz, ‘‘Linear Operators,’’ Part 1, Interscience, New York, 1958. 4. W. Hildebrand, ‘‘Core and Equilibria of a Large Economy,’’ Princeton Univ. Press, Princeton, NJ, 1974. 5. M. Ali Khan and M. Majumdar, Weak sequential convergence in L1Ž m , X . and an approximate version of Fatou’s Lemma, J. Math. Anal. Appl. 114 Ž1986., 569]573.
FATOU’S LEMMA IN BANACH SPACES
753
6. M. Papageorgiou, On a paper by Khan and Majumdar, J. Math. Anal. Appl. 150 Ž1990., 574]578. 7. F. Pucci and G. Vitillaro, A representation theorem for Aumann integrals, J. Math. Anal. Appl. 102 Ž1984., 86]101. 8. D. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 Ž1977., 859]903. 9. N. C. Yannelis, Equilibria in non-cooperative models of competition, J. Econ. Theory 41 Ž1987., 96]111.
199, 748]753 Ž1996.
0172
Fatou’s Lemma in Reflexive Banach Spaces S. I. Suslov Russian Academy of Sciences, Siberian Branch, Institute of Mathematics, No¨ osibirsk, 630090, Russia Submitted by Richard M. Aron Received September 7, 1994
A version of an approximate Fatou Lemma for a uniformly integrable sequence of functions with values in a reflexive Banach space is proved. The usual assumption that this sequence is pointwisely dominated in norm by a real valued integrable function is omitted. Q 1996 Academic Press, Inc.
Fatou’s lemma in n-dimensions proved to be very useful in mathematical economics Žsee, e.g., w4x.. Recently, the work in economics with infinite dimensional spaces of commodities attracted attention to this lemma in infinite dimensions w2, 5, 6, 9x. In the present paper, we prove a version of such a lemma, an approximate undominated one in a reflexive Banach space, thus generalizing the result of w2x. Let ŽT, A, m . be the interval w0, 1x with Lebesque s-algebra A and Lebesgue measure m , X be a reflexive Banach space with the dual space X *, and f n4 be a uniformly integrable sequence from the Banach space L1ŽT, X . of Bochner integrable functions f : T ¬ X. For a sequence A n4 of non-empty subsets of X write: w y Ls A n [ x g X : x s weak lim x k , x k g A n k , k g N 4 . Here and in what follows we use the symbol w to indicate that the corresponding topological notion should be understood with respect to weak topology. Our main result is: THEOREM. The following inclusion holds w y Ls
HT f
n
: cl 748
0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
HT w y Ls f . n
FATOU’S LEMMA IN BANACH SPACES
749
Here cl stands for the norm closure and the integral of a multivalued function t ¬ w y Ls f nŽ t . is understood in the sense of Aumann, i.e., as the set
½H
T
h: H g L1 Ž T , X .
and
h Ž t . g w y Ls f n Ž t . a.e. on T
5
of integrals of all integrable selectors of this multivalued function. Since all f n are m-essentially separably valued, we can and shall assume throughout the proof that X is separable. We use the following standard notations: co represents the convex hull; B Ž r . [ x g X : < x < F r 4 for r ) 0; sŽ x*, A. [ sup² x*, x :: x g A4 for x* g X *, A ; X; gr F [ Ž t, x . : x g F Ž t .4 for multifunction F : T ¬ 2 X . We shall prove the theorem by a sequence of lemmas. For t g T write: L Ž t . [ w y Ls f n Ž t . ,
F Ž t . [ cl co L Ž t . ,
LEMMA 1. The multifunction F is measurable Ž see w8x for definition.. Proof. Let Fm n Ž t . [ w y cl Ž B Ž n . l f k Ž t . 4 kGm . ,
t g T.
Being weakly compact, every ball B Ž n. is Polish in its weak topology w3, Theorem V, 6.3x. Consequently, we can use the results of w8x and easily obtain that the sets gr Fm n , m, n g N are A = BŽ X .-measureable, where BŽ X . stands for the Borel s-algebra of subsets of X. Then, since weakly convergent sequences are bounded in X, we have: gr L s
D F gr Fm n .
ngN mgN
So gr L is a A = BŽ X .-measurable set, and hence the multifunction L has a Castaing representation w8x, easily yielding the result. For t g T, n g N, set rn Ž t . [ < f n Ž t . < q lim inf < f k Ž t . < q 1, Fn Ž t . [ B Ž rn Ž t . . l F Ž t . ,
t g T , n g N.
LEMMA 2. The sequence rn4 is uniformly integrable and FnŽ t . / 0u a.e. on T for e¨ ery n g N.
750
S. I. SUSLOV
Proof. This follows immediately since, by the scalar Fatou lemma, the function t ¬ lim inf < f k Ž t .< is integrable. For x* g X *, write
a n Ž x*, t . [ max 0, ² x*, f n Ž t . : y s Ž x*, Fn Ž t . . 4 . Note that, by Lemmas 1 and 2, the functions a nŽ x*, ?. are well-defined and measurable. LEMMA 3. The sequence a nŽ x*, ?. is con¨ ergent to zero in measure. Proof. Since the sequences rn4 and f n4 are bounded in L1ŽT, R. and L1ŽT, X . respectively, a standard by-contradiction argument yields the existence of a point t 0 g T, a positive number « , and a subsequence k n such that
a k nŽ x*, t 0 . G «
; n g N,
Ž 1.
sup < r k nŽ t 0 . < R - `,
Ž 2.
sup < f k nŽ t 0 . < X - `.
Ž 3.
ngN
ngN
Due to Ž2. and Ž3. we can assume, extracting a further subsequence if necessary, that r k nŽ t 0 . ª r , w
f k nŽ t 0 . ª x 0
Ž 4. Ž 5.
hold for some r g R and x 0 g X. Observe now, this being the key point of the proof, that Ž4. implies s Ž x*, Fk nŽ t 0 . . ª s Ž x*, B Ž r . l F Ž t 0 . . , which together with a trivial consequence of Ž5. ² x*, f k Ž t 0 . : ª ² x*, x 0 : n yields lim a k nŽ x*, t 0 . s max 0, ² x*, x 0 : y s Ž x*, B Ž r . l F Ž t 0 . 4 .
Ž 6.
The right-hand side of Ž6., however, is equal to zero because x 0 , being certainly in B Ž r ., belongs to F Ž t 0 . by definition. This contradiction with Ž1. completes the proof of the lemma.
FATOU’S LEMMA IN BANACH SPACES
751
LEMMA 4. If f n4 or a subsequence thereof con¨ erges weakly to f g L1ŽT, X ., then f Ž t . g cl co w y Ls f nŽ t .4 a.e. on T. Consequently, for e¨ ery a g w y Ls HT f n there exists a function f g L1ŽT, X . such that f Ž t . g F Ž t . a.e. on T and HT f s a. Proof. Clearly it suffices to prove the lemma when the sequence f n4 w itself converges weakly to f. So let f n ª f. By the previous lemma, the sequence a nŽ x*, ?. converges to zero in measure. Hence, by w3, Corollary III.6.13x, we can assume, extracting a subsequence if necessary, that
a n Ž x*, t . ª 0
a.e. on T ; x* g X *.
Ž 7.
For « G 0 and x* g X *, set H Ž x*, « . [ x g X :² x*, x : F « 4 . Then, by the definition of a nŽ x*, t ., we have f n Ž t . g H Ž x*, s Ž x*, Fn Ž t . . q a n Ž x*, t . .
;t g T
Ž 8.
Note now that rn4 , being uniformly integrable, is weakly relatively compact in L1ŽT, R., so that we can suppose, extracting a subsequence if w necessary, that rn ª r for some r g L1ŽT, R.. Consequently, by Mazur’s theorem, for each n g N there exist nonn l n s 1, such that negative numbers l ni , i s 1, . . . , k n , with Ý kis1 i rnX [
kn
L1
l ni rnqi ª r ,
Ý is1
f nX [
kn
L1
l ni f nqi ª f .
Ý is1
Moreover, we can suppose, extracting a subsequence if necessary, that rnX Ž t . ª r Ž t .
a.e. on T
Ž 9.
f nX Ž t . ª f Ž t .
a.e. on T
Ž 10 .
Then Ž8. yields f nX Ž t . g H Ž x*, s Ž x*, F Ž t . l B Ž rnX Ž t . . . q a nX Ž x*, t . . where
a nX
kn
Ž x*, t . [
Ý is1
l ni a nqi Ž x*, t . .
a.e. on T Ž 11 .
752
S. I. SUSLOV
But
a nX Ž x*, t . ª 0
a.e. on T
and s Ž x*, F Ž t . l B Ž rnX Ž t . . . ª s Ž x*, F Ž t . l B Ž r Ž t . . . by Ž7. and Ž9. respectively. So Ž10. and Ž11. imply f Ž t . g H Ž x*, s Ž x*, F Ž t . l B Ž r Ž t . . .
a.e. on T ; x* g X *
and consequently f Ž t. g
F H Ž xUn , s Ž xUn , F Ž t . l B Ž r Ž t . . .
a.e. on T
Ž 12 .
ngN
holds for some denumerable set xUn 4 ; X * separating the points of X. Note that the right-hand side of Ž12. is equal to F Ž t . l B Ž r Ž t .. because the latter set is weakly compact in X. Thus, f Ž t . g F Ž t . a.e. on T which, together with the evident relation HT f s w y lim HT f n s a, yields the result. Now, the next lemma completes the proof of our theorem. LEMMA 5.
It holds cl
H L s cl HT F. T
Proof. Observe that in proving Lemma 1 the A = BŽ X .-measurability of the set gr L was established. In addition, the sequence f n4 is uniformly integrable by our assumption. Then, arguing as in w1, Theorem Ax, we obtain an integrable selector of L and apply w7, Corollary 3.3x.
REFERENCES 1. Z. Artstein, A note on Fatou’s lemma in several dimensions, J. Math. Econ. 6 Ž1979., 277]282. 2. E. Balder, Fatou’s lemma in infinite dimensions, J. Math. Anal. Appl. 136 Ž1988., 450]465. 3. N. Dunford and J. T. Schwartz, ‘‘Linear Operators,’’ Part 1, Interscience, New York, 1958. 4. W. Hildebrand, ‘‘Core and Equilibria of a Large Economy,’’ Princeton Univ. Press, Princeton, NJ, 1974. 5. M. Ali Khan and M. Majumdar, Weak sequential convergence in L1Ž m , X . and an approximate version of Fatou’s Lemma, J. Math. Anal. Appl. 114 Ž1986., 569]573.
FATOU’S LEMMA IN BANACH SPACES
753
6. M. Papageorgiou, On a paper by Khan and Majumdar, J. Math. Anal. Appl. 150 Ž1990., 574]578. 7. F. Pucci and G. Vitillaro, A representation theorem for Aumann integrals, J. Math. Anal. Appl. 102 Ž1984., 86]101. 8. D. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 Ž1977., 859]903. 9. N. C. Yannelis, Equilibria in non-cooperative models of competition, J. Econ. Theory 41 Ž1987., 96]111.