Internat. J. Hath. & Hath. Scl.
VOL. 12 NO. 1 (1989) 175-192
175
MEASURABLE MULTIFUNCTIONS AND THEIR
APPLICATIONS TO CONVEX INTEGRAL FUNCTIONALS
NIKOLAOS $. PAPAGEORGIOU
University of California
1015 Department of Mathematics
Davis, California 95616
(Received June 6, 1988 and in revised form September 26, 1988)
The purpose of this paper is to establlsh some new properties of set valued
ABSTRACT.
measurable functions and of their sets of Integrable selectors and to use them to
study convex integral functlonals defined on Lebesgue-Bochner spaces.
In this process
we also obtain a characterization of separable dual Banach spaces using multlfunctlons
and we present some generalizations of the classical
"bang-bang" principle to infinite
dimensional linear control systems with time dependent control constraints.
AND PHRASES.
KEY WORD
selectlon,
Souslln
asurable multlfunctlon,
space,
Weak
Itsndon-Nikodym
Integrably
property,
set
bounded,
valued
measurable
conditional
expectation, infinite dimensional linear control systems, bang-bang principle, convex
Integrand,
normal
subgradlent,
subdlfferentlal,
conjugate,
effective
domain,
absolutely continuous functlonal, slngular functlonal, Yoslda-Hewltt theorem.
1980 AHS SUBJECT CLASSIFICATION CODES.
1.
Primary 28A45, 46GI0, 46E30, Secondary 54C60.
INTRODUCTION.
In the last decade the study of measurable set valued functions has been
developed
combines
variety
of
in
the
theoretlcal
direction
and
the
direction
of
Many mathematicians have contributed significant results in this area,
applications.
which
both
extensively,
challenging theoretical
fields,
like
mathematical economics.
optimization
problems with important applications in a
theory,
optimal
control,
statistics
and
In all those areas the systematic use of multtfuncttons has
allowed people to make significant progress and solve many problems.
With a series of recent papers
extend
the general
theory of
related theory of multimeasures.
[18]
/
[25] the author has started an effort to
Banach space valued
multifunctions and the closely
The present paper continues this effort and provides
some applications of the theoretical results obtained.
Briefly this paper is organized as follows. In the next section we establish our
notation and for the convenience of the reader we recall some basic definitions and
176
N.S. PAPAGEORGIOU
from
facts
general
the
of
theory
and
mmltlfunctions
the
In section 3 we have gathered some results,
integrands.
of
properties
the
of
the
properties,
pointwise
its
selectors
integrable
of
set
about
information
of
theory
mmltifunction
a
measurable
in which starting
structure
of
we
from
extract
conditional
its
Some other related
expectation and the properties of the underlying Banach space.
In section 4, we
of functional analytic nature are also included.
proceed to a detailed study of the properties of the set of integrable selectors of a
observations
multifunction
we
and
present
an
application
control
to
theory
("bang-bang" type
Finally in section 5, we use the results obtained earlier in order to study
results).
that appear often in problems of optimization,
functionals
convex integral
control and mathematical economics.
optimal
With this combination of theoretical and applied
results, we want to emphasize the importance and the versatility of the theory of
multifunctions and attract the interest of mathematicians from different areas.
PRELIMINARIES.
2.
(, Z) be a measurable space and X
Let
separable Banach space.
a
Throughout
this work we will be using the following notations:
A C X: nonempty, closed, (convex)}
Pf(c)
A C X: nonempty, (w-) compact, (convex)}
P(w)k(c)(X)
Also we will be using the following additional three pieces of notation.
Ae2X’{}.
IAI
By
o(.,A) the support
function of A
F:
A multifunction
function
f
there exist
nl
n
problem
measurable
of
In applications
the
most
is
widely
for all
the
If X is a Souslin space and
xX:xeF(m)}
e
f
n
By
a
selector
central
used
in
selection
of
the
F(.).
The
theory
theorem,
is
of
the
for Polish spaces and was later
Z we will denote the universal
Z.
a-field corresponding to
THEOREM 2.1 [30].
[30].
be
to
said
selectors
extended to Souslin spaces by Saint-Beuve
then there exist
xeX,
This definition is equivalent to saying that
following one which was first proved by Aumann [I]
GrF={(m,x)
and by d(.,A) the
is said to be measurable if for every
X s.t. f(m)eF(m) is
existence
multifunctions.
s.t.
by
("Castaing’s representation"-see Castaing- Valadier [5]).
f:
function
of
II all’
X measurable functions s.t. for every
n
F() --cl {f ()}
A
Pf(X)
sup.
sup (x ,a), x eX
i.e., o(x ,A)
d(x,F(m))is measurable.
m
IAI
we will denote the norm of A i.e.,
Let
X,
ZxB(X), where B(X)
F:+2X’{}
is a multifunction
is the Borel
o-field of X,
Z-measurable selectors of F(.) s.t. F() c cl{f n ()} nl
MEASURABLE MULTIFUNCTIONS AND CONVEX INTEGRAL FUNCTIONALS
(l,r-,t)
a) If
REMARKS.
177
u-flnlte complete measure space, then
is a
r.
Z
b) Recall that a Souslln space is always separable, but it need be metrlzable (for
c) If F(.) is closed valued
and measurable in the sense defined earlier then GrF
r.xB(X) (graph measurablity).
r. (i.e.r. is complete).
The converse is true if r
example a separable Banach space with the weak topology),
Let (fl,E,) be a u-flnlte measure space and
-a,e},
S
Using
F
:f() e F()
e
we can define a set valued integral for F(.) by
f()d():feSIF }.
{
F()d()
We say that
if it is measurable and
iF(m) eL+I.
is Integrably bounded then
S
Let X
C_ Lx
be nonempty.
F:1
Pf(X)
is integrably bounded
Using theorem 2.1 we can see that if F(.)
fF are both nonempty.
F and
We say that K is decomposable (also known as "convex
with respect to switching") if for all
’In [9],
if(
Aer. and all
(fl’ f2
e KxK,
XAfl+XACf2
e X.
theorem 3.1., Hiai-Umegakl proved that if K is closed and decomposable, then
there exists
F:
Pf(X)
+
K
measurable s.t.
F(.) is integrably bounded.
S
F-
Furthermore if K is bounded, then
Using this fact, tllai-Umegakl
set valued condltlonal expectation for F(.) as follows.
sub-u-fleld of r.
{Er’f:fSIF}.
K- cl
F:
let
Pf(X)
be measurable with
[9],
went on and defined as
Let Zo be a
S
F
Define
r. -decomposable and so there exists
o
Then K is
r.
E
F:&I
Z
Pf(X)
o
SEZOF.
K
-measurable s t
The multlfunctlon
E
OF(.)
is the
set valued condltlonal expectation of F(.).
(,ZIN) be a complete,
Next let
space.
Let
f:OxX
/
RU {+}, f+.
u-flnlte measure space and X a separable Banach
Following Rockafellar [28] we say that f(.,.) is
a normal Integrand if the followlng conditions are satisfied:
i)
(re,x)
f(,x)
is ZxB(X) -measurable
meO, x
f(m,x) is lower semlcontlnuous
(l.s.c)
Using the celebrated "Von Neumann projection theorem" (see Castalng
if) for all
/
Valadler
[5], theorem Ill
followlng.
23) we can show that the above two conditions are equivalent to the
Recall that if g:X
RU{+(R)}
then eplg
+ epif(,.) is graph measurable
i’) the multifunctlon
ii’) for all
R, eplf(,.)
Pf(XxR)
Note that ii’) immediately implies that
condition
x:R
i)
a
normal
X is measurable,
f(.,.)
f(,x(m))
eplf(m,.) is measurable.
integrand
is
then
is measurable.
Because of
superpositlonally measurable
normal integrands are the Caratheodory integrands.
i.e.
if
A well known example of
A normal Integrand f(. ,.) is said
N.S. PAPAGEORGIOU
178
,
to be convex, if for all
f(m,.)
is convex.
Let
Let us also recall some notions from convex analysis.
{x*X*
f(x)
(x*,y-x)
yeX}.
f(y)-f(x) for all
subdlfferential of f(.) at x.
Also we define
and this is known as the conjugate of f(.).
f,X*
We define
xeX}
by f* (x* )=sup {(x* ,x)-f(x)
The conjugate and the subdlfferentlal are
related by the Young-Fenchel equality, namely:
(x*,x).
EX.
f
This is called the
x*f(x) if and only if f* (x* )+f(x)
Since we are dealing here with extended real valued functions we define the
{xeX:f(x)<}.
effective domain of f to be: domf
Finally, recall that X has the weak Randon-Nikodym property (WRNP) if it has the
Radon-Nikodym property for the Pettis integral.
MEASURABLE MULTIFUNCTIONS.
3.
We start with a result in which, knowing the structure of the set of integrable
of
selectors
a
we
multifunctlon,
deduce
properties
polntwise
some
of
the
Another such result was obtained by the author in [25] (theorem 5.1).
multlfunction.
Assume that
(H,Z,) is a complete, o-finite measure space and X a weakly
sequentially complete, separable Banach space.
THEOREM 3.1. If X* has the WRNP and S nonempty, bounded, closed and convex.
F
then
F(m)gPwkc(X)
(theorem 3.2)
From Hiai-Umegaki [9]
PROOF.
l-a.e.
that F(.)
know
we
Integrably
is
bounded and so for all mH’N,(N)--0, F(m) is bounded.
Suppose that for some e’N,F() is not w-compact.
{x
n
n)l
X
is
w-sequentially
complete
with no Cauchy subsequence.
Recalling that
{Xn}
and
theorem
the
fact
that
apply the result of Rosenthal [29] and deduce that
I_+ X
Then the Eberleln-Smulian
Ix n
n)l
n)l
give
us
a
sequence
is bounded, we can
is an
I -sequence.
Hence
a contradiction to the fact that X* has the WRNP (see Musial-Ryll
Na rdzewski 17 ).
REMARK. If X* has the RNP, then the result is immediate, since X is reflexive (see
But we know that in general the WRNP does
Diestel -Uhl [7], corollary II, p. 198).
not imply the RNP.
In fact, when X is a Banach lattice we can have a partial converse of the above
So assume that
(H,r.,) is a complete, o-finite measure space and X a
theorem.
separable Banach lattice.
THEOREM 3.2.
> F()Pwkc(X)
compact
PROOF.
S
C L
F
{/
A
If the following implication holds: "S
I
We will show that
(H) is nonempty
f(m)d(m):
feSIF }, Aer..
convex and w-
nonempty
F
-a.e.",then X* is separable and w-sequentially complete.
i
X.
Suppose not.
convex and w-compact then
Then M(.) is a
Pwkc( I)
is
Then from our hypothesis if
F(m)ePwkc(1)
u-a.e.
Set M(A)
-valued multimeasure with a
MEASURABLE MULTIFUNCTIONS AND CONVEX INTEGRAL FUNCTIONALS
Pwkc( I)
179
-valued density, a contradiction to example 2 of Coste [6].
So
gl X and
[7], p.95) we deduce that X has the
[7], we deduce that
Tzafrlrl
theorem I, c.4 of IXndenstrauss
then from a result of Lotz (see Diestel-Uhl
.
Since X is separable from corollary 8, p.98 of Diestel-Uhl
RNP.
X* is separable.
Also c
X* and so
o
[16] tells us that X* is w-sequentlally complete.
,
Also we have a weak compactness result for the set of integrable selectors of a
Another such result can be found in [25].
multlfunctlon.
property of decomposable subsets of
[7], Theorem 4, p. I04). For the convenience
Let (,E,) be a o-flnlte measure space
we recall the result here.
(Proposition 5.1
of the reader
But first we will need a
which was proved by the author in [26]
see also Diestel-Uhl
and X a Banach space.
PROPOSITION 3.1 [26].
l__f K C_ LX
decomposable and bounded then K is
is nonempty
uniformly Integrabl e.
Now assume that in addition X is weakly sequentially complete.
lf K C_ LX is nonempty, decomposable, bounded and w-closed, with no
then K is w-compact.
THEOREM 3.3.
1-sequence
From Proposition 4.1 we know that K is uniformly Integrable. Also since X
(see Talagrand [31]).
Combining these
facts with Corollary 8 of Bourgain [4] and the Eberlein-Smulian theorem we get that K
PROOF.
is
weakly sequentially complete, so is
Q.E.D.
is w-compact.
REMARK.
bounded.
X
If
is
separable
So indeed our result is a
K
then
S
F
w-compactness
where
F:R
Pf(X)
integrably
is
result for the set of integrable
selectors of a mmltlfunctlon.
4.
PROPERTIES OF THE SET OF INTEGRABLE SELECTORS.
This section is devoted to a detailed study of the properties of the set of
Those
integrable selectors of measurable (or graph measurable) multlfunctlons.
results are then applied to the analysis of a family of infinite dimensional llnear
control systema with time dependent control constraints. "The material of this section
will also be used in the next section, in the study of convex integral functlonals.
We will start with an auxilliary result that we will need in what follows and
which also generallzes Theorem 1.5 of Hlai-Umegakl [9].
(,Z,N)
Assume that
is a complete,
o-flnlte measure space and X a separable
Banach space.
LEMMA A.
PROOF.
we can flnd
exists
’{4}
lf F:
Clearly
fS
u(.) e
cony
I----- F
conv
LX, w*
S
C
F
s.t. f
[L
c0nv F
cony F
F
Suppose that the inclusion is strict so
is graph measurable and S
s.t.
But from [23] we know that:
S
cony
nv SF
F
then
S
S
From the strong separation theorem there
o(u,convS)
<
<u,f>
180
N.S. PAPAGEORGIOU
o(u,convSF1)
o(u,S F)
heS
f
(u(t),x)dv()
SUp
xeF()
< f
Hence o(u,convS F)
(u( m) h( ) )d u( 0)
]
O(u(to),F(o))dB()
u(0),f())dB()
.)eSlc--onvF
On the other hand since f(
==> f
o(u(m),F())-a.e
(u(),f())
f
sup.
we have
f()econ--’- F()B-a.e.
f
(u(),f())dB()
o(u(),F(m))dB()
Q.E.D.
a contradiction.
Using this lemma we can have the following Lyapunov type result.
(ll,r.,,) is a complete,
that
So we assume
o-flnite, nonatomlc measure space and X a separable
Banach space.
’{@}
THEOREM 4.1. If F:
is graph measurable and S
slc--nvF
Note that
_--[w
S
F
cony
g
let
neighborhood of g.
ueLv
SF,
So
and
e>0.
L(h)
fl,f2K
g-- Z
i-I
lifl,flSF.
<Uk,
Let
h
L:
m
R
k)mk-I ,e) be
V(g {u
Let
is convex.
and consider
A
We claim that
L(S)
m(A)=L(XA(fl-f2) ).
A L(XA(fl-f2))+L(f2
AeZ
So indeed
L(XAfI+XACf2) C
AUE L(KAfI+ XAC f2
L(S F) is convex.
Thus we can find
feSF
s.t.
L(S F) and
is convex.
It is easy to see that m(.) is a
,-continuous.
So applying Lyapunov’s
But since K is decomposable.
U
weak
>k=l
theorem we get that Range(m) is
==>
a
be defined by
vector measure of bounded variation which is
convex
F
(4 1)
F
Clearly L(.) is a continuous linear operator.
Let
cony
{he l<uk,g-h>[<e,k-I .....m}
V
where
cony
S
F
S (lemma a)
n
Next
S
Hence we immediately have:
is a closed, convex set.
C_ S convF
then
#
).
(heYe w indicates the weak topology on
PROOF.
F
L(fl),L(f2)
U
AeZ
L(XAfI+xAcf2).
MEASURABLE MULTIFUNCTIONS AND CONVEX INTEGRAL FUNCTIONALS
181
n
m
1%1 <uk’fl > }kffil
L(g)
L(f).
t;1
Therefore
V(g,{uk} 1’
_-T-w
S
F
}0s
S
cony
and so.
F
(4.2)
F
Combining (4.1) and (4.2) we finally have:
_--T-w
S
F
=conv S
.E.D.
F
An immediate important consequence of theorem 4.1 is the following result.
spaces remain as before,
w
_._If F:R
THEORM 4.2.
/
Pwkc(X)
From Benamara [2] we know that
PROOF.
SF, S
is measurable and
then
extF
S
extF
extF(to) is graph measurable.
to
The
SF
So
w
SlextF SlconvextF SFI
applying Theorem 4.1 we get that
(Krein-Mllman theorem)
Another interesting consequence is the following result about the set valued
Let
conditional expectation.
E -at oms
O
THEOREM 4.3.
If
E
Pf(X)
F:ll
sub-o-fleld of E and assume that
be a
O
is integrably bounded and
S
F
(.) has no
is w-compact in
r.
the.n.
PROOF,
F(to)
E
is
-a.e. convex.
Slr.
From Theorem 4.1 we have that
E
W
is convex.
F
Since
o
in
(Eo).
E
Therefore S
EEoF
is convex
E
F(to)
EoF
.a.e. convex.
is
Now we examine the strong closure of SF.
So
let
(R,E,)
is
then
S
a
complete,
o-finlte measure space and X a separable Banach space.
2X’{}
THEOREM 4.4. If F:R
PROOF.
Since
let L :R
n
-F
we get that S
is closed in
2X’{} be
Note that (to,x)
is graph measurable and
S
F
T.
F
S
F
F
defined by:
lx-f(to>ll
is a Caratheodory function and so it is Jointly
182
N.S. PAPAGEORGIOU
GrFeZxB(X).
f
n
:
/
{(,x)exX:llx-f(t)ll ’ 1_.}
n
Hence
measurable.
n)l, GrL eZxB(X).
Therefore for all
()
f()"
e,
n)1.
lfn()ll II f()ll
Since
Also by hypothesis
Apply Theorem 2.1 to find
n
f ()eL () for all
n
n
X measurable s.t.
’ fn
eZ.xB(X).
Then clearly for all
+ I, applying
the
dominated
convergence theorem, we get that
f
--" f
n
fESIF
=->
==>
S
SF.
F
Q:E.D.
However before passing to it, we need
The result has an interesting consequence.
It generalizes a similar result of Klai-Umegaki [9], who
[9]).
The
to have the following lemma.
required the multifunctlons to be closed valued (see Corollary 1.2 of
spaces remain as before.
LEMMA
B.
F2(m)
F I()
PROOF.
mA.
(where E
A
graph
2X’{o} be
g:A
Apply theorem 2.1. to find
R() for all
{n}n)l
Let
FA.
(A)>0 s.t.
FI(m)’F2().
defined by R()
/
SFI
and
measurable
Then there exists AE with
Suppose not.
Let R:A
E A).
all
g()
FI,F2:2X’{} are
l__[f
-a.e.
S
then
F2
FI()’F2() # 0
GrRCZAXB(X)
for
Then
X measurable s.t.
be a Z-partition of
Define
n
mn
{C
Then clearly
is a
mn m,n)l
>
find m,n)l s.t. (C
mn
O.
>
0 we can
mn
Then because of the decomposability of
F
produceslthe
(A)
mn
f() for
This
Since
Set
g() for
where f(.)gS
E-partltlon of A.
derived contradiction,
SFI
f eS
F1
while f
S
F
q2.E.D.
Now we are ready for the theorem.
THEOREM 4.5.
l_f
then
PROOF.
that F()
F:il
F()
X’{#}
2
Pf(X)
is
graph measurable and S F is nonempty and closed
-a.e.
From Theorem 4.4 we have that S F
F() -a.e. >F(.) is valued -a.e.
S
F
SF.
Apply
.
1emma 6% to
get
Now we will apply the results of this section to obtain versions of the "bang-bang
principle
for
infinite
control constraints.
dimensional
linear
control
systems
with
time
dependent
So consider the following system:
(t)
A(t)x(t)+B(t)u(t)
x(o)
x
0
u(t)eU(t)a.e
(4.3)
1
u(.)eL,
MEASURABLE MULTIFUNCTIONS AND CONVEX INTEGRAL FUNCTIONALS
tcR+
183
B(.)9oc(R+;L(X)),
A(t) is an unbounded, linear operator and
where
L(X) is the space of all bounded linear operators from X into itself. We will assume
that the linear evolution problem (t) -A(t)x(t), x(o)=x admits a fundamental
O
L(X).
solution $: {(t,s) :0st}
Conditions on A(.) that guarantee the existence
of $(..) can be found in Kato [12]. Then if
(R sX) we knoe that (4.3)
__OC
Here
B(.)u(.)
has a mild solution x
U
(.)Cx(R+)
given by:
t
x (t)
U
@(t,o)xO +
f
(t,s)B(s)u(s)ds,
+
O
Let R(t) be the set of the attainable points of system (4.3) using all feaslble
controls and let R (t) be the set of attalnable points of (4.3) using extremal
e
controls (i.e. u(t)eextU(t) a.e.).
The next
as
viewed
the relation between those two sets and can be
theorem establishes
an
infinite
generalization
dimensional
of
principle" (see Hermes- LaSalle [8]).
4.e.
_then for all
PROOF.
Clearly we
S
R(t) w
tR+ Re(t)
need to show
definition there exists u(.)eS
the
classical
extU
*
0
convex
R(t) C R (t)
that
So
e
let xcR(t)o
U s.t.
t
x(t)f(t,o)x O +
f
O
From Theorem 4.2 we know that there exists net
Then for every x*X* we have:
t
t
f (x*,$(t,s)B(S)Ub(S))ds
O
O
from the pcopectte off 0(,,,) (eee to
Hence s
+
B*(s)(t,s)xt belongs
LX
in
(.)S-
xtU
s.t. ub
IIx*lle have
[0,tl and so we have:
t
t
f
0
!
tL
f
’ t1 *<,)11 [12])
Note that
o
(B*(s)@*(t,s)x*,
f
Ub(S))ds
(B*(s)@(t,s)x*,u(s))ds
0
t
==>
f
(x*,#(t,s)S(s)u(s))ds
O
t
==>
f @(t,s)B(S)Ub(S)ds
f
O
O
(t, s)B(S)Ub(S)ds
t
Set x
b
Then
=> Xb
x.
Clearly
@(t,o)x +
f
$(t,s)S(s)u(s)ds
O
XbCRe(t) ->
x(t) cR
e
"bang-bang
(t)w==>
R (t)
e
w= R(t)w.
By
N.S. PAPAGEORGIOU
184
The convexity of the set is clear from the convexity of the values of U(.).
lot(R+)-
With an L
u(.), we
Q.E.D.
boundedness hypothesis on the control constraint multifunctlon
can improve the conclusion of Theorem 4.6.
OC
_--_-then
tR+,
for all
Re (t)wePwkc (X)"
R(t)
t
f
PROOF.
By definition R(t)--(t,o)x +
(t,s)B(s)U(s)ds.
o
o
Note that for all x*eX*
o(x*,(t,s)B(s)U(s))
Caratheodory
function
O(B*(s)*(t,s)x*,U(s))
Pwkc(X)-valued
and measurable, (s z*
from flxX* into
R.
Also since U(.) is
Therefore
it
o(z*,u(s))
is
is a
Jointly measurable and
Invoking theorem 111-37 of Castalno(B*(s)C*(t,s)x*,U(s)) is measurable.
(t,s)B(s)U(s) is a Pwkc(X)-valued, integrably
Valadier [5], we conclude that s
So we can apply proposition 3.1 of [18] (see also [22]) and
bounded multlfunctlon.
so s
get that
.
t
o
(t,s)B(s)U(s)dSPwkc(X) ==> R(t)Pwke(X) teR+.
E..D.
When X is a finite dlmensllonal Banach space, we obtain an extension of LaSalle’s
"bang-bang principle" (see Hermes-LaSalle [8]),to linear control syste.ms with time
2X’{} is graph
Recall that if F:R
nonconvex control constraints.
and (.)is nonatomlc, then fF is convex Csee Kleln-Thompson [13]
measurable, S
dependent,
F1
Theorem 17.1.6 and for a generalization to Banach spaces [23]).
THEOREM 4.8. __If U:
R+ Pf(X)
then for all
5.
is measurable and
tcR+, Re(t)
IU(.)t eLloc(1 R+)
RCt)ePkc(X).
CONVEX INTEGRAL FUNCTIONALS ON LEBESGUE-BOCHNER SPACES.
In this section we use the theoretical results obtained previously, to conduct a
study of convex integral functlonals which are defined on Lebesgue-Bochner spaces.
Our
work
in
this
section
extends
,
earlier
of
results
Rockafellar
[27]
(finite
dimensions) and Bismut [3] (finite dimensional or" separable, reflexive Banach spaces).
It is well known that if X is finite dimensional space and an integral functional
is weakly
lower semicontlnuous on
then the integrand is automatically convex in
the state variable (See Bismut [3], Theorem
and Rockafellar [27] Theorem I). Here we
extend this result to separable Banach spaces and we present a different, simpler
proof using Theorem 4.1.
First we need a Lemma.
Asse that (i,l,) is a complete, o-flnlte, nonatomlc
measure space and X a separable Banach space.
LEMMA
A. If F:
2X’{}
is graph measurable and S
140,
F
MASURABLE MULTIFUNCTIONS AND CONVEX INTEGRAL FUNCTIONALS
the_._n SF
PROOF.
have S
F(w)
.
F(w)ePfc(X)
First asstne
w-a.e.
that S
w-closed in
Then from Theorem 4.1
F
Using Lemma B of section 4 we conclude that
S
F
is w-closed if an only if
convF
=->
convF(w) -a.eo
Now assume that
F( w)
ePfc (X)
F()Pfc(X
-a.e.
Then S
F is closed and convex
Now we are ready for our theorem The spaces are as above
R is an integrand for x: X measurable we
Also if f:xX
set If(x)
f(m,x())d() (if the integral is not defined then we set
I
I) there exist
then f(w,.) is
PROOF
Let E:
Since the Integrand
(Xo(.)
that
let
(Xb’)v
’{@}
W-_xx
RtA
f
A
l(z)
I (x)
I(z)
4 llm
is w-l.s.c.
A
If(xb)
So indeed S
that E(w) e
E
Pfc(X)
4
-+).
If(x o) <
",
-lower semicontinuous,
-a.e convex
be the multlfunctlon deflnted by E()
f
We claim that SE is w-closed in
xR().
Then for all Ar. we have:
f( ’xb())d(w)
If(XAZ +
cf f(m’Xo(m))d() < "
So z +
s.t.
LX,
/0.
(x,A).
If(Xb)
and
If(x)
eplf(,.).
f(.,.) is normal E(.) is closed valued and measurable. Mso note
f+(.,Xo(.))e
Note that
Xo()L x
Is,(
If(.)
So it is w-
R is a normal integrand s.t.
I__f f:xX
b)
we
-ae.
closed.
THEOREM 5.1.
185
f (m)d(m)
A
XAcxo) Acf(a,Xo(W))dt(w
Also z()
/
(XAZ +
XAcXo)
is afflne continuous
Hence
A(w) d(w)-=>
f(w,x(w))
4
A(w) -a.eo
A
is w-closed in
__xR
W’a.e--> f(w,.) is
==> (x,A)S
Applying Lemma A we get
-a.e. convex.
Q.E.D.
Now we pass to the subdifferential of f.(.).
So assume that (,r.,) is a
complete, o-finite, measure space and X a separable Banach space.
We will need the
This result was first proved by oslda-Hewitt [32]
decomposition theorem for [Lx]
for X=R (see also Rockafellar [27]). Then it was extended to separable Banach spaces
by Ioffe-Levin [I0] and Rockafellar [28] and later to nonseparable Banach space by
186
N.S. PAPAGEORGIOU
Levln [15].
to
,
A functional
u(.)[Lx]
geL1,
if there exists
X
<
A
functional
{a
exist
In+
ill)
<
(g()), x())d() for all x(.)L
veiLX]
is
be
to
said
singular
X-
with
respect
if
to
there
c Z s.t.
n
i)
absolutely continuous with respect
*
>
u,x
is said to be
s.t.
nl
Rn nl
_c
v,x
>
(Rn
0 for all xL
[15]
THEOREM 5.2.
.
I.__
continuous on L
X
at
0 for all
s.t. x
where
R
Ar., (A)<(R) and
for some nl.
n
Yo
admits a unique decomposition
is absolutely continuous and
is singular with
Ys
IlYll l[Yoll + llYoll"
Furthermore
TttEORII 5.3.
X
A)
0
Every functional
Ya+ Ys
Y
respect to
li)
R is a convex, norl integrand and
f:l!xX
If(.)
is strongly
Xo(.)
.then 8If(x o)
,
c L
X
is nonempty and
*
W
Furthermore if X * is separable
th.en
w(LI,X
*
,Lx)-compact.
W
f(,x o()) is a
Pwkc(X)-valued
integrably bounded ltifunction.
.
,
and let x
*
-Xla+Xls be its decomposition according to Theorem
From Levin [15], Theorem 6.4, we know that:
5.2.
(If)
Since
Let
,
Let x ell x]
PROOF.
If(.)
x*eSIf(Xo )"
is s-contlnuous at x
O
if x*
S
8If(x o)
#
.
Then by definition we have:
<ta +
Note
If,(Xa*) + o(x;,domIf)
(x)
that
x, Xo>
because
of
If(Xo)+Clf)*Cx*)
the
continuity
IfCXo)+If, CXa)+oCx,domIf)
of
If(.)
at x ,x
0 then
< X’s, Xo > < o(x;, domlf)
which when used back in (4.4) produces a contradiction.
Therefore
x*s
0
==>
x*
X*a ==> If(xo)_ _c LX* ,
W
O
O
intdom
4.4)
If
and
so
MEASURABLE MULTIFUNCTIONS AND CONVEX INTEGRAL FUNCTIONALS
Xo(.)
187
Also the continuity at
tells
But the restriction of the w*-topology on
w(
*
,,
w
Now, if X*,
*
is separable, then
w
Also from Rockafellar [28]
*
is w(
If(x o)
Therefore
Lx)-topology.
81f(Xo is w*-compact
uslthat
L, coincides with the
in [L
x]
,,Lx)-Compact.
w
, LX,
we know that
(see Ionescu-Tulcea [1 1] ).
If(x o)
S
o
8f(,Xo ())
proposition 5.1 of [25], we conclude that
So applying
f(. ,x (.))"
Pwkc(X)-valued
is
Q.E.D.
Our result generalizes Theorem 2 and its corollary in Blsmut [3]. In this
integrably bounded multifunctlon.
REMARK.
paper X was assumed to be separable, reflexive.
Next we look at some special type of subgradlents namely extremal subgradlents.
This spaces are as above.
Xo(.), _-__-then extSIf(Xo
continuous on L
X at
u(0)
ext)f(o,x o(o)
.
is w(
If(xo)
extIf(xo
theorem we have that
extslsf( ,xo()) But
Hence u:Sel’xt)f(.
,Xo(.) -->
we
functional
Assigns
our
turn
w
SSf(. ,x o (.) ==> extIf.(xo)
[2], extS )f(
slext )f(
’Xo())
,x 0
)"
8f(O,Xo(O) )l-a.e.
Q. E .D.
the
for
of
conjugate
convex
the
integral
If: LXI/
that (R,Z,)is
THEOREM 5.5.
space and X a
nonatomlc measure
finite,
/
RU{+m} is w-lnf-compact, if for all
If f:RxX + R is a convex, normal integrand which is w-inf compact in
x for all eR and there exists
then.. If,(.)
is
topology).
Since
[15] or Rockafellar
the
x(.)eL.*.,A
If(x(.)) <
+ =,
(Here m(.,.) denotes the Macke y
w-lnf-compact,
from
Moreau’ s
, If
is
Since
from convex analysis we know that
is m-contlnuous at O.
(If)
[28]).
interior
s.t.
m(Lx,(R) w*’Lxl)-cntlnuus
hypothesis
by
Laurent [14]) we have that
in
complete,
a
Recall that f:X
A} is w-compact.
ER,{xX:f(x)
continuous
So by the Krein-Mllman
Lx)-Compact.
If(x o)
Also
u(o) Eext
attention
separable Banach space.
PROOF.
,,
*
from Benamara
Now
and for all
strongly
P-a.e.
We know that
PROOF.
If(.) is
uextSIf(Xo),
is a convex, normal integrand and
THEOREM 5.4. If f:xX
of
its
domain,
Also
[If]
If*
we have to show that
theorem
(see Levln
If,(.)is mIf,(.) is finite
eve rywhe r e.
From the fact that
If,(.)
is m-contlnuous at
O, we get that there
(see
exists
188
N.S. PAPAGEORGIOU
{filter of
VCNm(0)
we have
m(Lx
L )-neighborhoods of the origin} s.t. for all x*cV
.
+
If,(0)
If,(x*)
From the definition of the Mackey topology, we know that V
relatively w-compact set W in
{x*(.):Lx,w*
V
Since W is relatively
f
sup
ueW
eoaet
=.
cause
n
in
sup
f(x*(),u())d,() <
u
with
f*(,x*())dB(m)
Ak
If,(.)
fII
is m-continuous at
() <
f*(,X
A
f
O,
f
ugW
0 there exists 6
>
0 s.t.
I, for
.....
en
n.
()x*())d()-
ff*C,o)d(
k
AC
f*(,o)d()
<
.
k
<
l.
and
If*(’XAk(m)x*C))d"(O
--> f
fo all k=l
sup
(x*(t),u())du(m)
uW A
k
R
e
,
f
(XAk()x*(t), u())d,(m)
Therefore
XAkX*(.) V so:
Also sup
>
z"
k=
since
from Theorem
SO for all e
n
m).f
1}
(
(&i,Z,) Is finite, nonatoc, from Saks lem we knn that we can
{ }k= E
find
en
,
(x*(t),u(t))d(t)
we have that W is unifoly integrable.
Take e
is the polar set of a
So we can write:
f*(o,x*(o))d(o)
<
If, (0)
If, (XAkX*)
+
+(R).
==> If,(x*) <
and since
x*gLx,
was arbitrary, we have that dora
w*
Now we will obtain a description for
I f, =Lx,
domlf,
Q. E .D.
w*
So assume that (R,E,)is a
complete, o-flnlte measure space and X a separable Banach space.
We recall that if f:X
f:X
RU{+ }, f
RU{+ } is defined by
fo(h)
addition lower semi continuous, then
#
+
is convex, then the recession function
su{f(x+h)-f(x): xedomf}.
f(h)
sup
>0
f(x+%h)-f(x)
If f(.) is in
MEASURABLE MULTIFUNCTIONS AND CONVEX INTEGRAL FUNCTIONALS
THEOREM 5.6.
x
and
then we have
on
R is a convex, normal integrand, there exist
If f:xX
x*Lx,
If(x)<
s.t.
189
and
If,(x*)
and
for all
S,.(.,.)and
domIf,
<+"
If(.)
is lower semfcontfnuous
x*(.)sextdomlf,
we
x*()extdomf*(,.) -a.e.
have
PROOF.
Since by hypothesis
If(.)
is l.s.c, on
i
L
and convex, from Theorem 6.8.5.
of Laurent [14], we have:
(If)(R)
o(x,dom(If)*).
(x)
Since by hypothesis
domlf
, (If)*
If,,
while from a simple application of
the monotone convergence theorem (see also Blsmut [3], Proposition I) we have that:
(If).(x)
Therefore
If.
f f(,x(0)d()
Observe that domf*(,.)
U
o(x,d-mlf,).
But
n}
{x*X*:f*(,x*)
f(R)(0,x())
=->
o(x(,dom----f*(, .) ).
domf*(,.)
is graph
nl
measurable.
mso
So applying Theorem 2.2 of Hial-Umegakl [9] we get
x*Sdom,(.,.).
that:
,
sup (x* ,x(o) )d I()
x* domf*
j’Cx* (),xC ))d ()
sup
x*(. )dS-do,
C. ,.)
==> o(x,domlf,)
o(x,S
domf*(., .)
Since both sets are clearly convex, we conclude that:
domlf,
&do*(.,.)
The second part of the conclusion,
follows from the following equalities (see
Benamara [2]
extdoml f,
extS-do, (.,)
Sext-omf, (.,.)
Q.E.D.
190
N.S. PAPAGEORGIOU
AO(NCLEDGEMENT.
This research was supported by N.S.F. Grants D.M.S. -8602313 and
D.M.S.-8802688.
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