433
Internat. J. Math. & Math. Sci.
(1987) 433-442
Vol. i0 No. 3
CONVERGENCE THEOREMS FOR BANACH SPACE
VALUED INTEGRABLE MULTIFUNCTIONS
NIKOLAOS S. PAPAGEORGIOU
Department of Mathematics
University of California
Davis, California 95616
(Received May 21, 1986)
In this work we generalize a result of Kato on the pointwise behavior of a
P
LX(fi) (I _< p _< (R)). Then we
ABSTRACT.
weakly convergent sequence in the Lebesgue-Bochner spaces
use that result to prove Fatou’s type lemmata and dominated convergence theorems for
Analogous con-
the Aumann integral of Banach space valued measurable multifunctions.
vergence
results
are
also
for
proved
the
of
sets
selectors
integrable
of
those
In the process of proving those convergence theorems we make some
multifunctions.
useful observations concerning the Kuratowski-Mosco convergence of sets.
KEY WORDS AND PHRASES.
Convergence, measurable multifunctions, nonatomic.
1980 AMS SUBJECT CLASSIFICATION CODE.
1.
28A45, 46GI0.
INTRODUCTION.
In [I] Schmeidler motivated from problems in mathematical economics, proved a set
A different
valued version of Fatou’s lemma, for multlfunctions taking values in
n
proof and some additional results in this direction were obtained later by Hildenbrand
and Mertens [2].
Finally Artstein in [3]
provided the sharpest version of that result.
However
all the above authors apparently were unaware of an earlier analogous result of Kato
[4], for Banach space valued functions. The purpose of this note is to significantly
extend the result of Kato [4], use that extension to prove a Fatou’s lemma for Banach
valued’
space
multifunctions,
Hildenbrand-Mertens
extending
this
theorem for Banach space valued multifunctions.
results
way
the
works
of
Schmeidler
[I
[2] and Artstein [3] and finally prove a dominated convergence
for the sets
Then we obtain analogous convergence
of Bochner integrable selectors of the multlfunctions.
Our
results can have important applications in optimization, optimal control, differential
inclusions, abstract evolution equations and mathematical economics.
2.
PRELIMINARIES.
Let
(,l,)
space, with
X
be a complete,
o-flnite
being its topological dual.
P (X)
f(c)
{A X
measure space and
X
a separable Banach
We will use the following notations:
nonempty, closed, (convex)}
N.S. PAPAGEORGIOU
434
{A =-X
P (x)
wk(c)
nonempty, w-compact, (convex)}
acA
A i.e.
function of
for all
A i.e.
function from
for all
the support
is said to be measurable if it satisfies any of
Pf(X)
F
A multifunction
OA(’)
and by
aA
sup(x ,a).
aA
,OA(X
X
x
inf[[x-a[l
dA(X)
X,
x
the following equivalent conditions:
i)
for all
ii)
s.t.
there
o-field
measurable
of
functions
xX
F()}
x
ExB(X),
B(X)
where
is the Borel
(graph measurability).
X
of
{(,x)
{fn ")}n>l
(Castaing’s representation).
for all
GrF
iti)
is measurable
sequence
a
exists
cl{fn()}n>
F()
(x)
F()
X, m *d
x
We denote by
S
() i.e. SF
Bochner space
F(-) that belong to the LebesgueIt is easy to see
f() F()u-a.e.}
the set of all selectors of
F
(G)
{f(.)
that this set is closed and it is nonempty if and only if
L+. We say
F(’)I L+.
inf
xeF(m)
that
F
Pf(X)
X
SF,
Using the set
F(m)dM(m)
If
{
{A
nn>l
s
and
bounded
integrably
is
it is measurable and
if
F(-)
we can define a set valued integral for
S.
f(’)
f(m)du()
This integral is known as
{x s X
{x
x
w-
n+-lim An
We
say that the
X
A’s
n
lira
x
s
Xk,
x
k
Aumann’s integral.
X, we define
are nonempty subsets of
lira A
as follows:
E
Ank, k _>
A
converge to
>
x s A,n
lira x
I}
I}.
(denoted by
in the Kuratowski-Mosco sense
_K A) if and only if w- lira A
For more details we refer to
lira A
s
A
n
n
n
the nice works of Mosco [5], [6] and of Salinetti and Wets [7], [8] and [9].
A
3.
CONVERGENCE RESULTS FOR THE AUMANN INTEGRAL.
In this section our goal is to prove a Fatou’s lemma and a dominated convergence
theorem for the Aumann integral.
the
w-lira
PROPOSITION 3.1.
If for all
then for all
PROOF.
We start with an interesting observation concerning
Fix
*
x
a subsequence of {x
e X*
n n>l
n
x* e X *
An
lira o
and let
s.t. (x ,x
>I
g
x
n
k)
An
X
Assume that
of a sequence of nonempty sets.
and
*
(X)
o
An
-=
where
Pk (x)
G
w
*
(x.__._)
w-lira A
A n s.t. (x* ,x n
lira o
is a Banach space.
G
A
n
(x)
as k
-O
.
A
(x).
Since
{Xk}k>
Let
{x
n n>l
---G,
be
CONVERGENCE THEOREMS FOR BANACH SPACE
435
invoking the Eberlein-Smulian theorem and by passing to a subsequence if necessary, we
may assume that
Then
xew-lim
w
x
x.
k
(x* ,x)
An =>
<
*
(x)
o
=>
w-lira A
lira o
n
An
*
(x)
<
o
(x)
w-lira A
n
Q.E.D.
This leads us to the following interesting theorem that generalizes significantly
an earlier result of Kato
<
p
<
(,E,V)
THEOREM 3.1.
X
is a measure space,
{f (’)
n
If
e G()-a.e.
then
f(’)}
where
G() e
f() e
cony
a Banach space and
p
w-L
f() e
cony
n>k
,
x
e X
<
f())
<
and
{fn()}n>l-a.e.
w-lira
k
>
f ()v-a.e.
n
>
k
we have:
,
o
o (x)
cony
=>
p
Pwk(X),-a.e.
Then for all
(x
<
___x_+ f(.)
() fn (")
n>l
From Mazur’s lemma we know that for all
PROOF.
Let
.
and the sequence of vector valued functions was uniformly bounded.
Here
fn ()
[4], who had X to be reflexive with a uniformly convex dual,
(x ,f())
fn ()
n>k
<
,
fn ()
sup(x
n>_k
(x)
n>k
lira o
lim (x ,f ())
n
f ())-a.e.
n
(x) ,-a.e.
{f ()}
n
Using proposition 3.1 we get that
,
<
{fn()}
,
lim o (x)
=>
(x ,f())
<
o
0
-a.e.
(x)
w-lim
{fn()}n>l
(x)
w-lim {f
=>
f(m) e
cony
-a. e.
()}
n
n>l
w-li--- {fn ()} n>l -a.e.
Q.E.D.
Having this theorem we can have the
w-lim
version of
Fatou’s lemma for the
Aumann integral.
Now
(,l,)
is
separable Banach space.
a
nonatomic,
o-finite,
complete measure space and
X
a
N.S. PAPAGEORGIOU
436
THEOREM 3.2.
n
_>
Fn
I,
()
w-lira F ()
n
Let x
f
w-li---
E
s.t.
know that there exist
f
x
(")
SF n
(fl)
conv
is graph measurable and
cl
x
f
w-li--’
f
Fn ()d(m)
f
cl
integrably
w-li---
F ()dB().
for
s.t.
all
and
bounded
Then there exist
f fl fk ()d()"
s.t. x
k
k
[I0].
fk (’)
w-li---
(")
w-li--- Fn ()d(),
cl
multlfunctions
is
_
From the definition of the Aumann integral, we
x.
from theorem 3.1 we know that f()
>
Pwkc(X)
n
may assne that
F ()-a.e.
n
measurable
F ()d (m).
xu __w_+
fk
latter is w-compact in
we
are
G
f Fn()d()
w-li---
f1 Fnk()d()
quence,
Pf(X)_
where
is measurable
then
PROOF.
Fn
If
G()-a.e.
=-"
But
S
F
n
k
S
and the
G
So by passing if necessary to a further subse-
w-LIx_+
cony
f(")
E
w-lira {f
Fn()d().
S
G.
()}
n
Hence
f
x
n> l-a.e. ->
Since by hypothesis
f(m)d(m).
f() e
cony
w-li-
But
w-lira
F n ()
is nonatomic, we have that
cony
w-li-- F n ()d().
Thus finally we have that
which proves Fatou’s lemma for the weak llndt superior.
Q.E.D.
Next we w-Ill prove the s-lim version of Fatou’s lemma.
under less restrictive hypotheses on the sequence
Here
n
n>l"
is a complete, u-finite measure space and
(,l,)
This can be achieved
{F (")}
X
a separable Banach
space
THEOREM 3.3.
__If
Fn
2X\{$}
are integrably bounded and
{IFn (’)l}n>l
is
uniformly integrable
then
PROOF.
S
s-lira F
Let
f
x e
s-lira F f(a)d(a)
f
s-lira
s-lira F f(to)d(a).
Now consider the multifunctions
n
f Fn(t)d().
Then
x
f( to )dl (to
with
f(’)
L ()
n
(f(()) <
+ !}. Because the function
F n ()
d (x)
ts ratheodo, it ts superpostttoually easurable and so
(,x)
()
m
d (f(m))
f(
Is a Caratheodory
d ((f()) s measurable. en (re,x)
()
()
F
{x : F (m)
d
llx-f(m)ll
I
)I
437
CONVERGENCE THEOREMS FOR BANACH SPACE
{(,x)
So
function and so jointly measurable.
xX
e
F
n
[Ix-f()[[ < I-}n
(f())
()
d
n
e
--n
F /t
n
GrF
n
Apply Aumann’s selection theorem to find
e ExB(X).
fn () e Ln()
e
for all
.
From the definition of
F ()
f()d()
x and x
=>
F ()d()
n
n
x
s-lira
X
n
s-lira__
.
n
f
measurable s.t.
Fn ()
[ll] we know that
F ()d().
Fatou’s
Hence
n
lemma follows.
Q.E.D.
From Kuratowski [II], we know that an equivalent definition of
REMARK.
(x e X
s-lim F () is: s-lim F ()
n
n
(,x)
Note that
closed set.
d
F
(x).
(a)
lim d
and then so is
n+
F
n
O}
s-lira F (m)
is a
and that
(x)
n
(m)
n
being Caratheodory it is jointly measurable
lim d
n+
n
(x)
()
F
{(,x)
Hence
xX
lira d
rr+a, F
(x)
O}
ZxB(X)
=>
n
=>
s-lim F () is measurable.
ZxB(X)
(’))
n
n
Combining the two Fatou’s lemmata we can have a dominated convergence theorem for
Gr(s-lim F
the Aumann integral.
(,Z,)
So assume that
is nonatomic, complete, o-finite
X
mesure space and
a
separable Banach space.
THEOREM 3.4.
G()-a.e.
If
n
Pf(X)_ are measurable
Pwkc(X) integrably bounded
_K-_M__+
Fn()d()
then
cl
f
multifunctlons s.t.
and
Fn
()
<=>
F()
w-lira F ()
n
F (m)
n
_K_M_+ F()-a.e.
F()d().
F ()
This follows from theorem 3.2 and 3.3 if we recall that
PROOF.
F()-a.e.
F
G
with
s-lim F (m)
n
and
F(")
is
n
closed
and
valued
measurable.
Q.E.D.
REMARK.
If we assume that F("
is convex valued (which is the case if the
Fn()dv(m) _K-_M__+
are) then we have that
F()dv() [I0].
case we can relax the nonatomlcity hypothesis on
F’s
Furthermore in this
V(’).
We will close this section with a dominated convergence theorem for the Hausdorff
metric
h(’,’)
on
Pf(X).
Let (li,Z,V) be a complete, o-finite
THEOREM 3.5.
If
F
n
is uniformly integrable and
ii
P (X)
f
measure space and X a separable Banach space.
are measurable multifunctlons,
F () __h_+ F()
n
in measure
{IFn (’)l}n>l
438
N.S. PAPAGEORGIOU
then cl
Recal 1 that h(c 1
PROOF.
IF ()1
convergence theorem [12]
we get
cl
f
n
F()d())
+ 0 as
F()d. (m).
cl
IF()I"
+
n
n
clf
f Fn(m)d (m),
F())<
h(F ()
Also
h_+
f Fn()d()
f
F()d (m))
Then
f h(Fn()
_< f h(Fn (m)
F(m))dB (m).
extended
dominated
the
using
F(to))di(a) + 0
=>
h(cl
f
F (a,)d(a)
".
Q..E.D.
4.
CONVERGENCE RESULTS FOR THE SETS OF INTEGRABLE SELECTORS.
In this section we prove analogous convergence theorems for the sets
As before we will start with two Fatou’s type theorems.
S
F
n
But first we need the
following auxiliary result about the Kuratowski-Mosco convergence of sets.
Here
X
is any Banach space.
PROPOSITION 4.1.
Let
all
x
e X
c__
e X*
x
cony
w-lira A
x
*
for all
w-lim A
n
then
PROOF.
l__f
,
An
(x)
< O A(X
A.
A
Then there exist
(x ,x) ffi> (x ,x)
(x ,x k)
lim o
<_
lim o
An
(x)
_< O A(X
s t. x_----+ x.
=>
x
So for
conv A.
Q.E.D.
Now we are ready for the first Fatou’s type convergence result.
be a complete, o-finite
THEOREM 4.1.
s.t.
__If
{IFn(’)l}n>l
Let
d (u)
+/-.
S
Fn
Fn
Pf(X)_
fl
s-i im F
u(’) e
I-ul
fS
=
s-li.__m
n
().
inf
fe S
n
f
are
X
measurable
F
Fn
Then we have:
Fn
(u())d().
d
s-lira__
multifunctlons
SF n
f lf(to)
n
So using Fatou’s lemma [12] we get that:
So let
a separable Banach space
is uniformly integrable and
then S
PROOF.
measure space and
u(m)
Idu) f
xF (
n
(fl,l,)
CONVERGENCE THEOREMS FOR BANACH SPACE
li---
li----
(u)
d
SFn
f
d
F
n
<
(u())d()
()
li---
439
d
F
n
(u())d().
()
But from theorem 2.2 (i) of Tsukada [13] we have that for all
<
(u())
lira d
=> f li---
s-lira F ()
F
n
_
=>
s-lira F ()
n
<
(u)
lira d
(u(o))di()
d
s-li____m
Fn()
s
(u)
d
S
s-lira F
Ss-1 im
Pfc(X)-a.e.
F
n
s
S
So
s-lira F
Pfc(LxI)
was arbitrary we can apply theorem 2.2 (li) of Tsukada
S
s-lim F
n
(u)
d
SF n
Note that
L()
<f
(u())
()
d
(u())
d
F ()
and since u(’)
s
[13] and conclude that
s-li___m S F
n
n
Q.E.D.
We have the analogous result for
and
X
(,E,)
remain the same.
THEOREM 4.2.
>
The assumptions on the spaces
w-lira.
__If
Fn
F (m) c_. G(m)-a.e.
n
w-lira F ()
n
are measurable multifunctions s.t. for all
Pfc(X)
Pwkc(X)
I
G
where
is integrably bounded and
is graph measurable
w-lim S
then
c
F
cony
n
w-lira
Fn()
then w-lira S
c S
If in addition
F
s
S
-li--
Pfc(X)
F
n
for all
s
w-lim F
From the Dinculeanu-Foias theorem [14], we know that
PROOF.
,
Let u(’) s L
X
Then we have:
w*
a
(u)
S
(u(),f())d()
sup
f(’)s S F a
Fn
n
(u(o) ,x)dl (ul)
sup
xF (m)
n
;
a
f/
(u(u) )di (u).
F()
n
(Li)*
N.S. PAPAGEORGIOU
440
Then using Fatou’s lemma we get that
li--
li---
(u)
SF n
I
F
(u(m))d(m)
()
n
F
and since
w-lira F
f
(m)
n
F
n
(u(m))d(m).
()
(u())
w-lira F (m)
n
is by hypothesis graph measurable we have that:
(u())dv()
F ()
0
w-li---
i
<
(u())
()
n
lim
m E
But from proposition 3.1 we know that for all
lim
<
o
S
n
w-lira
(u).
F
n
So finally we have that:
lim o
<
(u)
Since
this
is
(u).
o
SF n
S
w-lim F
,
u(-)
for every
true
n
w-lira S
F
C conv S
n
w-li--
If in addition
of course closed and so
w-li---
F (.)
is
n
w-lim S
Fn
F
c__
from proposition 4.1
L
X w
conclude that
we
n
Pf c (X)-valued
S1 -lim
F
S
then
w-lira F
is convex and
n
n
Q.E.D.
Combining theorems 4.1 and 4.2 we can have a dominated convergence theorem for
the sequence
n
_>
{SF n
n>l"
THEOREM 4.4. __If
(m) _c G(m)v-a.e.
Our assmptions on
Fn
Fn
Pfc(X)
l
where
G
(fl,r. ,V)
and
X
remain as before.
are measurable multifunctions s.t. for all
Pwkc(X)
is integrably bounded and
Fn (m)
__K-M-+
F()v-a. e.
then S
PROOF.
F
K-M
n
S
F.
Note that because for all n
w-lira F () #
-a.e. But w-lira Fn ()
n
F()-a.e., we have that F()
_>
Fn () _c G()-a.e.
Pfc(X)-a.e.
is complete).
G()
with
Pwkc(X)
# -a.e. Also since s-lira F n ()
/ F() is measurable (recall (’)
and
F()
So using theorems 4.1 and 4.2, it is easy to see that
S
K-M_+
F
n
S
F.
Q.E.D.
We would like to have such a dominated convergence theorem for the Hausdorff mode
of convergence.
and
X
In this direction we have the following rusult.
remain as before.
The spaces
(,E,B)
441
CONVERGENCE THEOREMS FOR BANACH SPACE
THEOREM 4.5.
{IF n (.)I} nl
then F
I
Pfc(X)
PROOF.
F()
P (X) are measurable multifunctions s.t.
fc
integrable and F
F() in measure
n
n
is integrably bounded and
(Pfc(X),
First note that since
F()
that
F
If
uniformly
is
Pfc(X)-a’e"
Pfc(X)
for
()__h_,
S
h
F
h)
F
n
is a complete, metric space, we have
F(’) on a -null set we can have
and since (’)
is complete, the modified multi-
By
modifying
all
Also from the properties of the Hausdorff
function is still going to be measurable.
llFn()l-IF()II <_. h(Fn() F())-a.e. => lFn()l IF() in
by hypothesis {IFn )l}n>l is uniformly integrable, we deduce that
metric we have that
measure and since
IF(’)1
L+
F(’)
i.e.
S
F
So recalling that
L
X
SlF)
}n>I
,
f
lul I<i
(a).
and using Hormander’s formula we have that
i
(o
F
I0
sup._
as claimed by the theorem.
are convex, closed and bounded subsets of
sup
sup
f
bounde
is integrably
{S
Next note that
(u())
()
n
(x)-
(u()))dl()
ff
F()
(x){d()
O
n
h(F (), F())d().
n
{IFn(")l}n>l
Since by hypothesis
in measure then
f
h(F n (), F())du()
-->
0
h(S F
S
__h_ F()
Fn ()
is uniformly integrable and
0.
Q.E.D.
We will conclude our work with an important observation about the Kuratowski-Mosco
It is a very useful necessary condition for K- M
convergence of closed, convex sets.
convergence of such sets.
Assume that
X
THEOREM 4.6.
is a reflexive Banach space.
If
then A
{A
n n>l
C P
fc
(X)
[An[
sup
n>l
and for all
o
X
x
<"
(x)
A
An
and
O
n
PROOF.
Then
BM(0)
w-compact.
subsequence
suplAnl
n>
M
Let
is
and let
weakly--compac t
Let
x
w
Xk---+
n
x.
An
n
Then
>
x
and
I.
BM(0)
by
Then
w-lira A
A(X
_K-M_+
A
).
be the M-ball centered at the origin.
the
Eberlein-Smullan
c
{x
n n>l
A
=>
A
Bin(0
#.
theorem sequentially
and
Next fix
so we
can
,
x
X
find
and let
a
N.S. PAPAGEORGIOU
442
Xn
An
e
s.t. (x
A
,Xn
we can assume that (x*
lira o
o
An
An
).
On
(x)
_> OA(X
the
epi o
passing to an appropriate subsequence
By
lim o
k)
x
i.e.
, OA(’) ,
(’)
lim o
_< OA(X
(x)
An
*
(x).
*
(x)
A
hand
other
K-M
An
[6] and [7].
and
epi
w
Xk---
from
OA(’)
x
Mosco
A. Then (x *
[6]
x)
< OA(X
=>
that
know
we
{Xk}k>
and this implies that
So finally we have that
o
A
o
(")
n
A
(’).
Q.E.D.
REMARK.
Namely polntwlse con-
The converse of the above result is not true.
vergence of the support functions does not imply the Kurtowski-Mosco convergence of the
corresponding closed, convex sets.
and assume that
{x}
verge to
,
,
(x)+ (x
o
x
n
in the
K
x)
(x).
O
,
Here is a counter example.
but it does not converge strongly.
x
M
sense.
n
finite dimensional or otherwise
{x
C X
n n>l-{x
do not con-
Let
So
On the other hand for every
,
x
, ,
n
E
X
(x
x
n
So in corollary 2E of [I0], it mmst be added that
X
is
the result is not true as the previous counterexample
ilustrated.
AO(NOWLEDGEMENT.
This
work was done while
Department of the University of Pavia, Italy.
N.S.F. Grant DMS-8403135, DMS-8602313.
the author was visiting the Mathematics
Support was provided by C.N.R. and by
REFERENCES
I.
2.
SCHMEIDLER, D. Fatou’s Lemma in Several Dimensions, Proc. Amer. Math. Soc. 24
(1970) 300-306.
HILDENBRAND, W. and MERTENS, J.F. On Fatou’s Lemma in Several Dimensions, Z.
(1971), 151-155.
A Note on Fatou’s Lemma in Several Dimensions, J. Math.
Wahrsch. verw. Gebiete 17
3.
4.
5.
6.
7.
8.
Economics 6
ARTSTEIN, Z.
(1979), 277-282.
KATO, T. Accretive Operators and Nonlinear Evolution Equations in Banach Spaces
in the Pr.oceedings of the SEmposium on Nonllnear Functional Analysls, ed. F.
Browder, AMS, Providence, RI, 1968, 138-161.
MOSCO, U. Convergence of Convex Sets and of solutions of Variational Inequalities, Advances in Math. 3 (1969), 510-585.
MOSCO, U. On the Continuity of the Young-Fenchel Transform, J. Math. Anal. Appl.
35 (1971), 518-535.
SALINETTI, G. and WETS, R. On the Relations Between Two Types of Convergence for
For Convex Functions, J. Math. Anal. Appl. 60 (1977), 211-226.
SALINETTI, G. and WETS, R. On the Convergence of Sequences of Convex Sets in
Finite Dimensions, SlAM Review 21
9
(1979), 18-33.
SALINETTI, G. and WETS, R.
On the Convergence of Closed-valued Measurable Multifunctions, Trans. Amer. Math. Soc. 266 (1981), 275-289.
Set Valued Operators, Trans. Amer. Math.
I0.
PAPAGEORGIOU, N.S. Representations
Soc. 292 (1986), 557-572.
II.
KURATOWSKI, K. Topology I, Academic Press, New York, (1966).
ASH, R. Real AnalTsis and Probability, Academic Press, New York (1972).
TSUKADA, M. Convergence of Best Approximations in a Smooth Banach Space, J.
Approx. Th. 40 (1984), 301-309.
IONESCU-TULCEA, A. and C. Topics in the Theory of Lifting, Springer, Berlin (1969).
12.
13.
14.
of