Advances in MATHEMATICAL ECONOMICS pot

Advances in
MATHEMATICAL
ECONOMICS
Managing Editors
Shigeo Kusuoka
University of Tokyo
Tokyo, JAPAN
Akira Yamazaki
Meisei University
Tokyo, JAPAN
Editors
Robert Anderson
University of California,
Berkeley
Berkeley, U.S.A.
Charles Castaing
Universite Montpellier II
Montpellier, FRANCE
Frank H. Clarke
Universite de Lyon I
Villeurbanne, FRANCE
Egbert Dierker
University of Vienna
Vienna, AUSTRIA
Darrell Duffie
Stanford University
Stanford, U.S.A.
Lawrence C. Evans
University of California,
Berkeley

Berkeley, U.S.A.
Takao Fujimoto
Fukuoka University
Fukuoka, JAPAN
Jean-Michel Grandmont
CREST-CNRS
Malakoff,
FRANCE
Norimichi Hirano
Yokohama National
University
Yokohama, JAPAN
Leonid Hurwicz
University of Minnesota
MinneapoUs, U.S.A.
Tatsuro Ichiishi
Hitotsubashi University
Tokyo, JAPAN
Alexander loffe
Israel Institute of
Technology
Haifa, ISRAEL
Seiichi Iwamoto
Kyushu University
Fukuoka, JAPAN
Kazuya Kamiya
University of Tokyo
Tokyo, JAPAN
Kunio Kawamata
Keio University

Tokyo, JAPAN
Norio Kikuchi
Keio University
Yokohama, JAPAN
Tom M aniyama
Keio University
Tokyo, JAPAN
Hiroshi M atano
University of Tokyo
Tokyo, JAPAN
Kazuo Nishimura
Kyoto University
Kyoto, JAPAN
Marcel K. Richter
University of Minnesota
Minneapolis, U.S.A.
Yoichiro Takahashi
Kyoto University
Kyoto, JAPAN
Michel Valadier
Universite Montpellier II
Montpellier, FRANCE
Makoto Yano
Keio University
Tokyo, JAPAN
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S. Kusuoka, A. Yamazaki (Eds.)
Advances in

Mathematical Economics
Volume 10
Springer
Shigeo Kusuoka
Professor
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Akira Yamazaki
Professor
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Meisei University
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Table of Contents
Research Articles
C.
Castaing, M. Saadoune
Komlos type convergence for random variables and random
sets with applications to minimization problems 1
J P.
Gamier, K. Nishimura, A. Venditti
Capital-labor substitution and indeterminacy in
continuous-time two-sector models 31
T. Ibaraki,
W.
Takahashi
Weak and strong convergence theorems for new resolvents of
maximal monotone operators in Banach spaces 51
S. Iwamoto
Golden optimal policy in calculus of variation and dynamic
programming 65
S. Kusuoka
A remark on law invariant convex risk measures 91
A. Rubinchik, S. Weber
Existence and uniqueness of
an
equilibrium in a model of
spatial electoral competition with entry 101
H. Hata,
J.
Sekine
Publisher's
Errata:

Solving long term optimal investment
problems with Cox-IngersoU-Ross interest rates 121
Subject Index 123
Instructions for Authors 125
Adv. Math. Econ. 10, 1-29 (2007) Advances
in
MATHEMATICAL
ECONOMICS
©Springer 2007
Komlos type convergence for random
variables and random sets with applications
to minimization problems
C.
Castaing^ and M. Saadoune^
^ Departement
de
Mathematiques, Universite Montpellier
II,
34095
Montpellier Cedex
5,
France
(e-mail: )
^ Departement de Mathematiques, Universite Ibnou Zohr, Lot. Addalha, B.P. 8106,
Agadir, Maroc
(e-mail: )
Received: August 10, 2006
Revised: October 16, 2006
JEL classification: C61
Mathematics Subject Classification (2000): 49J40, 49J45,46N10

Abstract. Let £ be
a
separable super reflexive Banach space and let (^, T,
\x)
be a com-
plete probability space. We state some Komlos type theorems in the space
C?^
iSl, T, \x)
of ^-valued random variables and a version of Komlos slice theorem in the space
£^ , ,^x(^, T, /x) of convex weakly compact random sets. Weak Komlos type the-
orems for some unbounded sequences in £jr(^, T,
^x)
and £j.,[F](^, T, /i) when
F is a separable Banach space are also stated. A Fatou type lemma in Mathematical
Economics and minimization problems on convex and closed in measure subsets of
£^ (^, T,
/JL)
are presented. Further Minimization problems and Min-Max type results
involving saddle-points and Young measures are also investigated.
Key words: Biting Lemma, Komlos convergence, minimization, Min-Max, saddle
points, young measures
1.
Introduction and preliminaries
Throughout £" is a separable Banach space,
{Q,J^,
/x) is a complete probabil-
ity space, >C^(^, T, jx) is the space of all ^-measurable £-valued functions
defined on Q. Let cwk(E) be the set of all nonempty convex weakly com-
pact subsets of E. Let us denote by >C^^^^^^(^, ^, /x) the space of all scalarly
^-measurable cw;/:(£')-valued mapping defined on Q (alias convex weakly

2 C. Castaing and
M.
Saadoune
compact random
sets).
Recall that
a
cu;/:(£')-valued mapping X : Q -^ cwk(E)
is scalarly ^-measurable if the support functions
5*
(jc*,
X
(.))
are
^-measurable
for all X* G E\ Given a convex weakly compact random set X, we de-
note by |X| the ^-measurable real valued function |X| : Q, -^ \X((JO)\ :=
sup{|5*(x*,X(ft>))| : 11^*11 <
1}.
We
refer to
[18]
for details concerning Convex
Analysis and Measurable Multifunctions. Also we will use the following Um-
iting notions. Let
(Cn)neNu{oo}
be a sequence of nonempty closed convex sub-
sets of E,
{Cn)nef^
Mosco

Converges
to Coo if
the
two following inclusions are
satisfied:
Coo C s-liCn := {x e E : \\x -
XnW
->
0;
Xn
e Cn}
w-lsCn
:={x e E \Xn^^ X weakly; x„^ e C„ J c Coo-
Given two nonempty subsets B and C in £", the gap between B and C is
defined by
D{B, C) = M{\\x -y\\:x eB,y eC}.
The slice topology
Xs
on cc{E) (nonempty closed convex subsets of E) is the
weakest topology r on cc(E) such that for each nonempty bounded closed
convex subset B of E, the function C i-> D{B, C) is r-continuous.
(Cn)neN
slice
converges
to Coo if one has
lim D(5, Cn) = D{B, Coo)
for all nonempty bounded closed convex subset 5 of £". It is well-known that
the sUce convergence and the Mosco convergence coincide on cc{E) when E is
reflexive. We refer to [6] for the topologies on closed convex subsets in Banach
spaces. If F is

a
reflexive Banach space, any bounded sequence (/«) in the space
£}^(Q,
!F,
ix) has the Mazur property, namely, there exist a subsequence (g^)
of ifn) and /oo e
C\{Q.,
T, \x) and a sequence
i}in)
of convex combinations of
{gm).
i-e. hn
G
co{gm : m > n},Wn E N, such that (hn) converges a.e. to /oo,
with respect to the norm topology of F. Further if F is super reflexive, then any
bounded sequence (fn) in £^(Q, T, ji) has the Komlosproperty, namely, there
exist a subsequence (fa(n)) in
C\^(^,
T, /i) and /oo e £]^(Q, T, fx), such that
lim -Ey^i/y^o) = /oo
a.e., with respect
to
the norm
topology,
for every subsequence
(/^(n))
of (/«(«))•
Let
(/„)«eNu{oo}
in

C\(Q.,
T, \x) , the notation /x-lim«_^oo fn = /oo means
that (fn) converges to /oo in measure. For more on Komlos theorem [20] in
L}^(^, T,
jx)
where F is super reflexive (or B-convex) space, we refer to [7,
19].
Komlos type convergence with applications 3
In § 2 we present several versions of Komlos theorem [20]. The first ones
concerns with a version of Komlos theorem in £^(^, J", /x) and a version of
Komlos sHce type theorem on the space >C^u;^(£)(^, -^^
M)
when £" is a super
reflexive. The second ones deals with weak Komlos type theorems for some un-
bounded sequences in
C\^{Q,
T, /x) and £^,[F](^, T, \i), respectively, here
fi\,\F\(Sl, T, \i) denotes the space of all F^-valued functions f : Q. -^ F'
such that for all jc e F, the scalar function {x, f) is integrable and such that
the function |/| : ^ ^ R given by \f\{(jo) := H/MHFS
(JO
e Q is integrable,
when F is a separable Banach space, here the notation weak Komlos means that
the associated Cesaro sums converges a.e to the functions under consideration,
with respect to the weak topology of F and the weak* topology of F\ respec-
tively. Our main purpose is to introduce a new type of tightness condition for
sequences in these spaces. Namely, a sequence (/„) in £)r(^, T, /x), is Mazur
tight if it satisfies the condition (*): for every subsequence (/„^) of (/„) there
exists a sequence (r„) in £^(^, T,
JJL)

with r„ e co{\\fni(.)\\
'•
i > n} such
that
Hm
sup„ r„ e £jj(^, ^, /i) and similarly for (gn) in £^,[F](Q,
!F,
/x). It
is worthy to mention that the above Mazur tightness condition does not im-
ply that (fn) is bounded in C]^(Q, T,
JJL).
Indeed, it suffices to consider the
space £jj(^, ^,
M)
where Q = [0, 1] endowed with the Lebesgue measure
and fn is given by fnico) := «^l[o,i/n](^),
<^
e ^. Then, (/„) is not bounded
in
C^^^iQ,
T,
IJL)
but satisfies (*), because it converges a.e. to 0. These con-
siderations led to several tightness conditions in the study of Fatou lemma in
Mathematical Economics [15,16] and allow to give a new light in the problem
under consideration. In § 3 a series of new apphcations illustrating the results
obtained in the previous sections is given.
We
present a characterization of con-
vex closed sets for the convergence in measure by means of Komlos theorem in

C^^(Q,
T, /x). In particular we show that for every convex and closed in mea-
sure subset
W
of .C^C^, T, /x) and for every convex weakly compact random
set r, the set
<Sr
H-
7i where
<Sr
is the set of all ^-measurable selections of F,
is convex and closed in measure. A Fatou-type lemma in Mathematical Eco-
nomics and some Minimization problems are also given. We provide also two
min-max type results involving Young measures and saddle points for a class
of convex integral functionals.
We refer to [7,14,19,20,23] for Komlos theorem for vector-valued random
variables and to [4,12,17,21] for random sets.
2.
Komlos theorem for random variables and random sets
with applications
The following version of Komlos theorem in £^(^, T, /x) is crucial for our
purpose.
4 C. Castaing and
M.
Saadoune
Theorem 2.1. Let E be a separable super
reflexive
Banach space and let (fn)
be a sequence in
C^^(Q^,

T,
[x)
satisfying the following
condition:
Of or
every
subsequence (fm^) of (fn) there exists a sequence (rn) in
C^(Q,T,
p) with
rn e co{||/„.(.)||
'.
i > n] such that limsup^
rnico)
is finite for each
CD
in Q.
Then there exist a subsequence (gn) of{fn) and an E-valued
random
vector f
such that
1 «
Z
hi{a))-
f{a))\\
=0, a.e.coe Q,
liml
" ",=1
for
every subsequence {hn)
of(gn).

Furthermore,
if we
suppose in this condition
lim sup„ r„ e
Cl^(Q,J^,
p), then f^o e
C\{Q,,
T, p).
Proof By hypothesis, there exists a sequence (r„) of the form
rn^Y.^1\\fi^n\\
iein
with
X^
>0 and ^.^j k^ = I such that Umsup„
rn{(o)
is finite (equivalently
sup„
rn(co)
is finite) for each a; in ^. For p e N, define
Ap :=
{(o
e Q
:
supr„(a;) < p}.
n
As Unip-^oo
l^(^p)
=
1.
we can choose pi € N such that p(Ap^) >

1 —
^. By
integrating we get
SUPX^? / \\fi+n(co)\\dp<pi
Hence, there exists a subsequence (f^) such that sup„ /^
11 /^^ (co) \
\dp < p\.
In view of the Biting lemma (see e.g. [13], Theorem
6.1.4)
there exists an
increasing sequence (5^) in T with lim^ p{B^) =
1
and a subsequence {g\) of
(/^^) such that {g\) is uniformly integrable on each
Ap^
H
B^. Now, by virtue of
Komlos theorem [19], there exist a subsequence of {g\) still denoted by {g\),
f^ e £^(^, T, p) and
cp]^
e
C\^{Q,
T, p) such that the following hold
lim ||-Ef^i/z/(a;) -
f^((o)\\
= 0, a.e.
co
e Ap,
n-^oo n
and

hm -Ef^il|/i,Mll =
(plc^
a.e.
co
e Ap,
n^oo n
for every subsequence Qin) of {g\). As {g\) is uniformly integrable on each
B^
n Ap,, by Lebesgue-VitaU theorem, we get
Komlos type convergence with applications
5
V^,VAG^,
lim /
-J:^^^\\hi\\dn=
f
(pldfji
for every subsequence (hn)
of (gl).
This
is
equivalent
to
V^,VAe^,
lim /
\\gl\\diJi=[ cp^dfji.
Next, applying condition
(*) to (gl)
instead
of (/„) and
using again

the
pre-
ceding arguments
we
find
a
measurable
set
Ap^ with iJiiAp^)
> 1
—
|, an
increasing sequence
(B^) in T
with Hm^
M(5^)
= 1, a
subsequence
(g^) of
(gl),
f^ €
£^(^, JF, /x) and(^^
e
/:]^(^, JT, /x) such that the following hold
lim ll-Sf^i/z/M
- /^Mll = 0,
a.e.
co
e Ap,
for every subsequence

Qin)
of
{g^)
and
V^,VAG^,
hm /
\\gl\\dfi=
[
cpl^dii.
""^"^JADAry^nB} JAOAn^nBi
Repeating the preceding arguments provides a measurable set
A^^
with /x (^^^)
>
1 —
^,
an increasing sequence
(BJ^)
in^withlim^
M(^^)
=
1,
a subsequence
(8n)
of
(g^^),
/4 e
£^(^,
^,
M)

and
^^
G
4(^, ^,
/x) such that the fol-
lowing hold
lim ||-Ef^i/z,M -
/4MII
= 0, a.e.
co
e Ap,
for every subsequence (hn)
of
(g^)
and
V^,VA€^,
lim/
\\gl\\dn=f cpldfjL.
Finally, define
^pi
•= ^Pi and A'^^ := Ap, \
Ap,_,
for/: > 1
k=oo k=oo
gn '•=
gl f^
:=
X
^^pj^
^^^ ^~

•=
S
^^W^^'
/:=1
k=\
Then
(
k=oo
\
/k=oo
\
6 C. Castaing and
M.
Saadoune
because /x(Ap^) > 1
—
^ for all k eN. Further, it is not difficult to see that
lim ll-Ef^i/i/M - foc((o)\\ = 0, a.e.
co
e Q (2.1.1)
n-^oo n
for every subsequence {hn) of
{gn)
and
V^, V/:, VA 6 ^, lim /
WgnWd^i
= f (pocdfi.(2A.2)
Now, let us prove the second part of Theorem 2.1. Applying the new
condition to the sequence (gn) provides a sequence (r„) of the form r„ =
Y.iein^'i^^^i+nW with X^ > 0 and ZieinK = 1 such that limsup^r^ e

£^(^, J^, /i). From (2.1.2) and Fatou lemma it follows that
/ (Pood 11= lim X^? / Wgi^nWd^i
- lim / Tx'lWgi^nWdn
-^^Pk^^q ieIn
L
limsupr^JjU. (2.1.3)
A'p.C^B^,
n^oc
Since the sequence (^Sf=i^n+/) is uniformly integrable on each A^^ fl 5^, it
follows from (2.1.1) and Lebesgue-VitaU theorem that
lim / |hEf^i^,+,|I^M= / WfocWdfi. (2.1.4)
Therefore, by (2.1.4), (2.1.2) and (2.1.3) we deduce that
/ ll/ooll^M<lim-I],ti / Wgn+iWdn
= lim / WgnWdfi
= / (Poodfi
J
A'
nfii
^Pk
<i
'-L
Hmsupr^^jLC.
p.^B^g
«^00
Komlos type convergence with applications 7
Then it follows that
/ ||/ocll^M = S^'T 1™ / WfooWdli
< E^ Hm /
Urn snip Kfidfi
= /

XmiswpVnd^i
< +00.
The proof
is
therefore complete. •
Before going further let us mention a useful convergence result.
Proposition 2.1. Let H be a subset of
C^^(Q,
T, /x) with € = 0, 1. Then the
following are equivalent:
(1)
Given any sequence
(fn)
in
7i,
there exist a
subsequence (gn) of(fn),
cpoo
i^
£^^(Q, T,
/JL)
and
an increasing
sequence (Q) in J-with limjt^oD
l^{Ck)
= 1
such that for every k eN, Ick^oo ^ >^^+(^, ^,
M)
and
VAeJT, lim / \gn\dfi=

(PoodfJi.
(2) Given any sequence (fn) in H there exists a sequence (rn) in C^(^, T, \x)
with rn e co{\fi\ : i > n} such that (rn) pointwisely converges a.e to a
measurable function roo ^ >Cj^(^, J^, ji).
(3) Given any sequence (fn) in H,
there exists
a sequence (rn) in C^(Q, T, /x)
with
rn
€ co{\fi
I
: / > «} such that hm sup„ r„ € £^(^, T,
/JL).
Proof Suppose (1) holds. Let (/„) be a sequence in H. There exist a subse-
quence (gn) of (fn),
(foo
e
C^-^
(^, T,
/JL)
and an increasing sequence (Ck) in
J^ with lim^-^00
M(Q)
= 1 such that
V/:, VA
G
^, lim /
\gn\d/ji
= / (foodfi < 00.
"^^"^JADCk JAnCk

For
each/:
choose
n^
such that, for every n > nk, J^
Ignldf^
< +oc and define
the following sequence of integrable functions:
gn,k = ^Ckgn ifw >nk,
= 0 otherwise.
It is clear that, for each k, the sequence
(\gn,k\)n
or(^\ L^)-converges to
1Q<^OO. By Lemma 3.1 in [17], there exists a sequence
(rn,k)n,k
with r„,^ ==
8 C. Castaing and
M.
Saadoune
ll'iLnK\8i,k\
where X^ > 0 and Xtn = 1 such that, for every k, {rn,k)n
converges a.e. to Ick^oo- Since
Wk,
WcoeCk,
lim y
XI
\gi,kI
= lim Y
A^
\gi |

= lim V
X^
|g,-1
/=n i=n i—n
and limits00 /xCQ) =
1
we deduce that the sequence (^{L^
A^
|g/1)
converges
a.etO(;^oo.Thisproves
(2).
The
implication (2) =^ (3)
is
trivial.
Now,
to
prove
the
implication (3) ^ (1) let (/„) be
a
sequence in H and let (g„),
(poo,
Apj^
andB^,
(k, q > 1), be defined as in the proof of Theorem
2.1.
Since lim^
M(5^)

= 1,
then there exists
qk
>
1
such that M(^^^) >
1
- p Taking Q := u|=^(Ap. 0
B^.) it is clear that (gn),
(Poo
and (Ck) have the required properties. D
There is a simple version of Theorem
2.1.
Corollary 2.1. L^r Ebea
separable
super
reflexive Banach
space and let (fn)
be a sequence in
C^^(Q,J^,
fi) such that
Um
sup„
11 /«(<^) 11
is
finite
for each
co
in ^. Then there exist a subsequence (gn) of (fn) and an E-valued random
vector f^ such that

1 "
\im\\-y^hi((o) -
foo{(o)\\
=0, a.e.
co
eQ,
n n ^^^
i—\
for every subsequence (hn) of
(gn).
/jf Umsup„
||/n(.)||
e £^(^, ^, /x), then
Proof Corollary
2.1
is a direct consequence of Theorem
2.1,
because for every
subsequence (/„^) and for every sequence (r„) in £^ with r„ €
co{|
|/„.
11
: / >
n] we have
lim
sup
rn
(co)
< lim sup 11 fn (co)
11,

co
e Q,
n n
see Lemma
4.1
in [15] or more generally Lenmia 1.1 in [5]. D
Remark.
Theorem 2.1 is not true for unbounded sequences, it suffices to take
Now we proceed with some significant variants of Theorem 2.1 when the
Banach space E is not super reflexive. We recall some tightness notions.
Let F be a separable Banach space. Let cwk(F) (respectively TZwc(F)) be
the set of all convex weakly compact (respectively, weakly closed, ball-weakly
compact) subsets in F; a weakly closed subset in F is ball-weakly compact if
its intersection with any closed ball of F is weakly compact. If F is reflexive,
any weakly closed subset of F is ball-weakly compact. Similarly, any weak*
Komlos type convergence with applications 9
closed subset of
the
weak* dual of
a
Banach space F, is ball-weakly* compact.
A sequence (/„) in
C]^{^,
T, /x) is 'JZwc(F)-tight, if, for every e > 0 there
exists a 7?.u;c(F)-valued measurable multifunction
Vg
: Q ^ F such that
sup fi{{(jo
eQ: fn{o)) ^ ^^(0))}) < £
n

Theorem 2.2. Let Fbea separable Banach
space
and D := {e'p)p>\bea dense
sequence in F' for
the Mackey
topology.
Let (fn) be a IZwciF)-tight sequence
in
C\
{Q,
T,
jJi) satisfying the condition
C):for
every subsequence
{fn,^)
of(fn)
there exists a sequence (rn) in £|^(^, T,
IJL)
with rn e co{||/n,(-)ll
^
' ^ «}
such that
lim sup„
r„ {co)
is finite for
each co in
Q.
Then there
exist a
subsequence

(gn) of(fn) and an F-valued
random
vector f^o such that
1 '^
V/7>1,
-^i^ep,hi(co)^->{^e'p,foc(co)y a.e.coeQ,
i = \
1 ""
Vw e
C%[F]{Q,
T, /x), /x- lim - V(w, hi) =
(M,
/OO>
i=\
for
every
subsequence (hn) of(gn).
Furthermore,
if we suppose in the condition
(*), Umsup„ rn(.) eC^^(Q, T, /x), then foo ^
C.\(Sl,
T, /x).
Proof Let
us
prove the
first
part of Theorem
2.2.
By Proposition
2.1,

there exist
a subsequence {gn) of (/„),
(poo
e >C^+(^, T, /x) and an increasing sequence
(Ck) in T with lim^t /x(Q) = 1 such that for every /: e N
lim / \\gn\\dii= / ipoodpi. (2.2.1)
for all
A
e ^. It is clear
that,
for each ^ >
1,
the sequence
(1Q
^„)
is uniformly
integrable and 7^u;c(F)-tight in C],(Q, T, /x). Using (2.2.1) and Corollary 2.1
in [23] via an appropriate diagonal procedure, we find a subsequence of {gn)
still denoted
{gn)
and /4
G
il\{Q.,T,
/x) such that
1 ""
^k > 1,
V/7
> 1, lim - Tie' IcM^)) = iC
fU^))^
a.e. oyeQ

1 ""
V/: > 1, Vi/ e
C%[F]{Q,
T.
M),
/X
- lim - Yd/,
l^/i/)
= («, /4)
for every subsequence {hn) of (g„). Putting
C; — Ci and C^ := Q \ U ^' ^r
i^
> 1
10 C. Castaing and
M.
Saadoune
and
foo •= 2Ld ^C[foo^
k=\
it is not difficult to verify that
k=OQ
1 "
Vp>l, ^J^^-Y^{e'p,hi{w)) = (e'p,fo^{co)), a.e. c; 6 J2 (2.2.2)
f=i
1 "
VM
e 4,[F](fi, T, fi), M- lim - Y{u, hi) = (u, f^o) (2.2.3)
i=\
for every subsequence (/z„) of (^„).
Now, let us prove the second part of Theorem 2.2. Applying the new

condition to the sequence {gn) provides a sequence (r„) of the form r„ =
Hiein KW^i^nW with X^ > 0 and ^.iein K = 1 such that limsup^ r„(.) e
C\^(Q,
T,
fji).
From (2.2.1) and Fatou lemma it follows that
< / limsupr„d/x. (2.2.4)
On the other hand, since the sequence
{\ckSn)
is uniformly integrable, from
(2.2.3) and Lebesgue-VitaU it follows that (2.2.5)
^k >
1,VM
e L^,[F](^,J^,/x), Um /
(M,
-S,tig/+itWM= / {u, foo)dii.
Since the norm ||.||^i is weakly lower semi-continuous, (2.2.5) impHes
V/: > 1, /
WfocWdfJi
< Uminf / ||-Ef^i^,+^||J/x
< Uminf / -Sf^illgz+itll^/x
= lim / WgnWdii
Jci
Komlos type convergence with applications 11
where the two last inequalities follow from (2.2.1). Thus the above estimate and
(2.2.4) imply that
VA:
> 1, / ll/ooll^M < / limsuprnd/ji.
Jc:
Jci n

Hence
/
||/OO||JM
= 5]+!? /
WfocWdfi
k
+CX)
< ^kS / limsupr„J/i6
Jq n
lim
sup
rndfi < +oc.
/
JQ
As a special case of Theorem 2.2 we give the following result.
Corollary 2.2. Let F be a
separable
Banach space and let (fn) be a sequence
in
C^iO,,
T, /x) satisfying the condition (*) of
Theorem
2.2 and the condition
(**).•
there exists
a measurable IZwc(F)-valued
multifunction
F \ Q =^ F such
that fn
(CL>)

e T{a})for alln e N and all
co
e
Q
Then
there exist a
subsequence
(gn) of(fn) and an E-valued
random
vector /o© such that
1 "
Vx* 6 F\ lim - V(x*, hi((o)) ->
(JC*,
/oo(<^)>, a.e.
co
e Q
/=!
for every subsequence Qin) of
(gn),
here the negligible set depend only the
subsequence under
consideration.
Furthermore,
if we suppose in the condition
(*) limsup^ rn{.) € £jj(^, J^, /x), then foo e C\{a, T, /x).
Proof It clear that (/„) is 7^M;c(F)-tight. Applying Theorems 2.1 and 2.2,
respectively, to the sequences (H/nlD^ and {{e[, fn))n (k > 1) via a diagonal
process provides a subsequence (gn) of (fn) and /o© e i2^(^,^, ^i), (or
e £jr(^, •^,M)»if we suppose in condition
(*),lim

sup„r„(.) e i2}^(Q,^,/x))
such that
lim -i:^_.\\hi(co)\\ exists, a.e. coeQ, (2.2.5)
1 "
Vk>h lim -y(4,/z/(a;)) = (4,/oo(^)), a.e.a;€Q, (2.2.6)
for every subsequence (hn) of (gn). To complete the proof take a subsequence
(/i^)of
(/;,)andputr(.) := /?(.)nr(.):^£, wherer(.) :^ sup„ ii:f^i||/z,(.)||.
12 C. Castaing and
M.
Saadoune
Then, by (2.2.5), r{co) is weakly compact a.e and Vn e N, ^DJLi/i/M e
coF
(co)
a.e. Hence, it follows from
(2.2.6),
by using
a
routine density argument,
that
1
""
VJC*
6 F', Um - Y(jc*, /z/(a;)) = (x\ /oo(^)), a.e. a; e Q.
/=1
Remark, Theorem 2.2 is valid in the space £^(Q, T, fi) under obvious modi-
fications.
Now we present some convergence properties for a class of unbounded
sequences in the space LJ^,[F](^, J", /x) of all F'-valued mappings / : ^ ->
F^ such that

co
h-> {f((o),x) is integrable,
VJC
€ F, and such that |/|(.) :=
||/(.)||/r/ belongs to L^(^, T, /x), when F is a separable Banach space. We
summarize some properties of this space. For more details, we refer to [5,18]
and the references therein. We will endow L\^,[F](Q, T, \i) with the norm
A^i(/) = A^i(ll/ll), /GL),,[F](Q,J^,/X)
1
here
A^i
denotes the norm
in
L^ (^, ^,
/x).
By Theorem
4.1
in
[5]
L^ (^, ^, /x)
1
is included in the topological dual (LL[F](^, T, [X))' of LL[F](/X) and we
have
A^i(/)= sup f {h,f)dix
heTii J^
here H\ denotes the set of all simple mappings from ^ into the closed unit
ball Bp of F, so that the mapping / i-^ N\(f) is lower semicontinuous on
L}„[F](Q, JT, /X) for the topology or(L},,[F](/x), Lf(fi)).
The following result deals with some convergence properties for a class
of unbounded sequences in (L^p,[F](Q, T, /x), A^i) and leads to interesting

apphcations in several problems of convergence of F'-valued scalarly integrable
random variables, in particular, Fatou type Lemma in
L\,\F\(Q.,T,
IJL).
Theorem 2.3. Let F be a separable Banach
space.
Let (fn) be a sequence in
(LJ^,[F](^,
T,
/X),
N\)
which satisfies
the
condition
{""): for
every subsequence
ifrik) ofifn) there exists a sequence (rn) in L^(^, ^, /x) with
rn G
co{\frn
\
:
/ > n} such that
Hm
sup„
r„ 6 L]^(Q, !F, fi). Then there exist a subsequence
(gn) ofifn), foQ G
L^p,[F](Sl,
T, /x) and an increasing sequence (Q) in T
with limjt /x(Q) =
1

^^^^ that
V/:> l,Vi;eL^(^,JP^,/x), lim / {v,gn)d[i= f (u,/oo)^M
Komlos type convergence with applications
13
and
such
that
fooico)
^colf^
w*-cl{fm((o)
: m > n} I, a.e.
Furthermore,
we
have
1
"
VJC
e F, lim - ^{x, hi) = (jc,
/o©),
a.e,
n^oo
n ^-^
i = \
for
every
subsequence
(hn)
of(gn), here the
negligible
set depends only

on
the
subsequence under consideration.
Proof.
Step 1 Applying Proposition 2.1
to the
sequence
{\fn\)
provides
a
sub-
sequence {gn)
of (/„),
(poo
e
L^+(^,
T, ii) and an
increasing sequence
(Q)
in
!F
with limjt
M(Q)
=
1 such that
for
every
k e N
lim
/

\\gn\\d^l=
I
(pocdfJi. (2.3.1)
for
all A e T.
Then
for
every
k e N,
(lck8n)neN
is
uniformly integrable
in
L]^f[F](Ck,
Ck
n T,
/X|Q).
In
view
of
Theorem
6.5.9 in [13] and a
diagonal
procedure,
we
produce
a
subsequence
(not
relabelled)

of
{gn)
and a
sequence
(/4) with
/4 e
LJ„[F](Q,
Q n T. /x|cj
such that
I p|u;*-d{/^(a;):m>/7}
I,
/4(c^)€co(
[
]u;*-d{/^(a;):m>/7}
|, a.e. a; E Q
(2.3.2)
and such that
V/:
€
N, Vu
€ Lf{n, T.
M),
lim /
(i;, gjJ/x
= / (u,
/4)^/x.
Define
C;
:= Ci and C^ := Q \ |J Q for
i^

> 1
and
/oo
:= Z lq/<
it
14 C. Castaing and
M.
Saadoune
It follows that
V/:6N,VMeL^[F](^,jr,/x),
lim /
{u,gn)dfjL=
[ (u,
foc)dfi.
(2.3.3)
On
the
other hand,
let D :=
{ep)p>\
be a
dense sequence
in F for
the norm
topology. Applying Theorem
2.1,
respectively, to the sequences ((^p, gn{')))n.
(p
€
N), and

(| |^„
(.)11)^
via a standard diagonal extraction procedure provides
a subsequence (not relabelled)
of
(gn)
such that
1
"
V/7
e N, lim
-^{ep,hi{a))) exists a.e. (2.3.4)
n-^oo
n ^^
and
1
"
lim -
V||/Z/(CD)||
exists a.e. (2.3.5)
n-^oo
n ^~^
i—\
for every subsequence (/z„)
of
(^„). By (2.3.5) the sequence
(^
^YH^x
ll^«ll)
i^

pointwise bounded almost everywhere.
By
(2.3.3)
and
(2.3.4)
it is
immediate
that
1
""
Vp
G
N, Hm -
X^^P' ^^•(^))
=
<^P' /^)-
Using the separability
of F
and the pointwise boundedness
of (^
X/^=i
I
l^n
11)
we get, by a routine argument
1
"
lim -y]{e,hi)
->
(^,/oc

for all ^
€ F
and almost everywhere.
Step 2 /oo
e
LUF]{Q,J=', /X).
AS
A^i(/)=sup
[
{Kf)dn=
f
\\fi.)\\dfi,
V/e
L^,[F](/x)
A^i
is
cr(Lj.,[F],
L^)
lower semicontinuous, (2.3.3) implies
V/:
> 1, /
ll/ooll^M
<
liminf
/
||gn(.)II^M
=
lim /
WgnWdfi
= /

(^oo^/x
Komlos type convergence with applications 15
where the two last inequahties follow from (2.3.1). Hence
/||/oc||^M = S+!? / WfocWdfJi
Jn Jcl
^ ^j£T / ^oodfJi = / (Poodfji < +00.
Finally by (2,3.2) we have
fooico) ^colf] w''-cl{fm{co) : m > n} I, a.e.
Remarks. If
(|
/„ |) is bounded, condition (*) is satified. See for instance Propo-
sition 2.1.
Now we proceed to a new version of Komlos sUce theorem in the space
C^cwk(E)^^' J", ix). Compare with
([17],
Theorem 4.1 and Corollary 4.2). See
[4,12] for other related results. Let us recall the following result.
Proposition 2.2. Hess [9] Let F be a separable Banach space, and D =
(^k^k>l
be a dense sequence in F' for the Mackey topology. Let (Xn) be a
sequence in ^^yjj^(f)i^^ ^)- Assume that the two following conditions are sat-
isfied:
(i) There is a convex weakly compact valued random set C such that, Vn >
l,^a)eQ,Xn{(o)cC(oj).
(ii) Wk > 1, lim„-^oo
^*(^^'
Xnico)) exists a.e.
Then there exists a convex weakly compact valued random set X^ and a neg-
ligible set N c Q such that
lim 8*(e\ Xn(oj)) = 8*(e\ X^oicv))

/i->oo
for all e' e F' and for allco e Q\N.
Theorem 2.4. Let E be a separable super reflexive Banach space. Let (Z„)
be a sequence of convex weakly compact random sets in E such that for every
subsequence (X„^) of (Xn) there exists a sequence (rn) in £^(^, T, jx) with
rn e co{\Xn.
\ '.
i >n} such that lim sup„ ^n(^) is finite for each
co
in Q. Then
there exist a subsequence (Xp(n)) of{Xn) and a convex weakly compact random
set Xoo such that, for all bounded closed convex subset B in E and for every
subsequence (Xy^n)) of(X^(n))> the following
hold:
lim D (B, -'^'j=iXy^j)(co)) = D(B, Xoc(co)) (*)
for almost every co e ^.
16 C. Castaing and
M.
Saadoune
Proof.
Let D[ := (^pA:>i be a dense sequence in
BE*
for the norm topology.
For each k, we pick a maximum ^-measurable selection
aj^
of Z„ associated
to el, that is, (^|, a^) = <5*(^^, X„). By virtue of Theorem 2.1, there are a
subsequence (Xy3(„)), a subsequence (<T^(„)) and a^ G £^(^, T, /x) and areal
valued ^-measurable function cp^ such that for each k the following hold
Um 5* (4, -E^^iX^(y)(a;)) = (p^ico) a.e. (2.4.1)

for every subsequence (Xj/(„)) of (X^(„)), and
1
•^n ^k
lim -£"=!<(.)M = a^ico) a.e. (2.4.2)
for every subsequence (o^y(„)) of (o^L^))- For simphcity we set
n -^
for all n e N and for all
tD
e Q. Again by Theorem
2.1,
we may assume that
for each
6t)
G
^,
sup 15nMl < +00 (2.4.3)
n
a.e. By (2.4.1), (2.4.3) and Proposition 2.2, there is a convex weakly random
set Xoo and a negligible set N such that
^lim^r L\ \i:^%xXyU){cJ\ = 5* {e\
Xooico))
(2.4.4)
for all
(oj,
e*) e (Q\N) x
BE*-
NOW
by obvious properties of (or;f), (2.4.1),
(2.4.2) and the definition of s-li
^

D^^j Xy(j)
(co)
we have
5*
«,
Xooico))
= [el a^(co)) < 5* I el s-li^Y.^yU)(co) j (2.4.5)
for all k and almost every a; G ^. By (2.4.4), (2.4.5) we conclude that (Sn)
Mosco-converges to Xoo- We will prove (*) that is a formulation of slice con-
vergence in
C^cwkiE)^^^
•^'
/^)-
^y (2-4.4) we have
Komlos type convergence with applications 17
liminf D(B,
Sn(a)))
= liminf sup
{-5*(JC*,
Sn(o)))
-
5*(-JC*,
B)]
> sup liminf{-(5*(jc*,5^(a;))-r(-Jc*,B)}
x*eBE'
• sup
x*eBE'
-limsup(5*(x*,
Snico))
-

8*(-x\
B)
= sup
{-(5*(jc*,Xoo(ft>))-5*(-;c*,5)}
= D(B,Xoo(co))
for any bounded closed convex subset B inE and for almost every
co
e Q, This
proves the liminf
part.
Let us prove now the limsuppart. Let (gk) be a Castaing's representation
of 5-//^Ey^jXy(j)(.). Then we have
Vo;
G
^, Vx e £:,
VA:
E
N, limsupJ Ix, -'E''^^Xy(j)((jo)] < d(x,gkia)))
which impHes that
limsup J Ix, -l^^-iXyn)(co) I < inf d(x, gk((o))
n-^oo \ n •' ) keN
=
dUs-ii^i:^^,Xyu)((^)Y
Now let 5 be a bounded closed convex subset of E, then
limsupD I 5, -J:''-^Xy(j)(a))] = limsup inf d(x, -T^''-^Xyn)(a)))
n->oo V ^ / n-^oo xeB n •'
< inf limsup J (x,
'-Ti^^-^Xy(j)(a))
I
xeB n^oo \ ^ /

< inf dlx^s-li-T^I.Xya^ico))
xeB \ n J~ J
^Di^B.s4i^-i:)^,Xy^j^{co)^
<D(B,Xoc{(o))
for any bounded closed convex subset B in E and for almost every
co
e ^. D
We give some apphcations of the preceding results.
Proposition 2.3. Let E be a separable super
reflexive Banach
space.
Let H be
a
convex
subset
ofC^^(Q,
T, /x).
Then
the following are
equivalent:
18 C. Castaing and
M.
Saadoune
(a)
H is
closed for the topology of
convergence
in
measure,
(b)

For any
almost
everywhere
pointwise bounded
sequence
(un) in H,
that
is,
SUp||M;^(6t>)||
< +00
n
for almost
every
CO
e
Q,
there
is
a
subsequence (uy^n))
andu^
e £^(^, T, p)
such that
/x
- lim
-E'J_iMy(/)
=
Moo
with respect to the norm topology.
Proof

(a) =^ (b). In
view
of
Theorem
2.1,
any
almost everywhere pointwise
bounded sequence
(un) in H has the
Komlos property. Hence there exist
a
subsequence
{uy(n))
and
MQO
€
>C^(^,
^, p)
such that
lim
-
H'J-i My
(/)=
Moo
n-^oo
n •^•~
a.e., with respect
to the
norm topology.
In

particular,
we
have
IX- lim -
E'J^i My
(;)=
Moo
with respect
to the
norm topology. Since
H is
convex
and
closed
for the
con-
vergence
in
measure,
it is
immediate that
MOO
(b)
=^ (a). Let
(M„)
be a
sequence
in H
which converges
in

measure
to
u
e C^^(Q,T, p).
There
is a
subsequence
of (u„) of
(M„)
which converges
to
M
for
almost every
co
e Q.
Thus
the
sequence
(i;„) is
pointwise bounded
for
almost every
co
e
Q. Using
(b)
and Theorem
2.1,
there is

a
subsequence (ua(n))
and
Voo
^H
such that
lim -Ey^jiJaQ-)
=
Uoo
a.e., with respect
to the
norm topology.
So we
have
u = v^o e
Hfoi almost
every
o)
eO •
The following result is
a
combined effort of Theorem 2.1 and Theorem
2.4.
So
we
omit
the proof.
Proposition 2.4.
Let
Ebea separable super

reflexive
Banach
space.
Let (Xn)
be
a
sequence
in
^lyj^E)^^'
^ ^^^^
^^^^
SUP|XJG4(^,J^,M).
Komlos type convergence with applications 19
Then there exist a subsequence {X^(n)) of{Xn) and a convex weakly compact
random
set Zoo such
that,
for all bounded closed
convex
subset B in E and for
every subsequence {Xy(n)) of(Xp(n)), the following
hold:
lim D (B, -^^-^Xy^j^ico)) = D(B, XocM) (*)
for almost every
6D
G
^.
For the sake of transparency, let us examine first the particular case where
Xn G L^(^, ^, /x). Since E is reflexive, by Dunford-Pettis theorem, (X„) is
relatively sequentially weakly compact in L^(J^, ^, /x). See [11] for further

references of weak compactness in Bochner and Pettis integration. So there
is a subsequence (X^) of (Xn) which converges weakly to an element
Y^Q^
6
L\(SI,T ,{i) with I Fool Sg and, by Mazur's lemma, there is a sequence (Z„)
of convex combinations of (Xn) which converges strongly to Foo for almost
every
o)
^Q Using Theorem 2.1 or Proposition 2.3, we get more. There is a
subsequence (Jn) of (Xn) and a random integrable E-valued vector Zoo such
that
lim||-5]f^iZ/M-XocM||=:0, a.e.a;G^, (*)
n n
for every subsequence (Z„) of (F„). Let W be the unit ball of L^(^, J^,
JJL).
Then (*) imphes
lim -Ef^i {W(a)), Ziioj)) =
{W{a)),
X^(co)),
a.e. a)eQ. (**)
n n
Integrating equahty (**) gives
lim-Ef_i f (W(aj),Zi(co))i^(dcv)= f
{W{cv),
Xoc(o)))fi(da)) (
)
for every subsequence (Z„) of (F„). (***) shows that (Yn) weakly converges
to Xoo in the Banach space L^(^, T, fi). Similarly, in the multivalued case.
Proposition 2.4 provides a shce type convergence for the sequence (Xn). This
result is sharper thanjhe Mazur's TL convergence initiated in [17], namely,

there is a sequence (Xn) with Xn e co{Xni : m > n] (i.e. X„_has the form
Xn = ^ZnK^i with 0 < Af < 1,
i:^^^k^
= 1) such that (Xn) converges
in the linear topology xi [6] to a convex weakly compact random set Zoo ^
^\wk{E)^^'
^) for almost everywhere a;
G
^, that is
VJC*
G E\ lim r (x*, Xn) = r (jc*. Zoo)
n->oo
Vx
G
E, lim d(x, Xn) = d(x, Zoo)
M—>00