Internat. J. Math. & Math. Sci.
Vol. 2 No. 2 (1979) 209-221
209
p INTEGRABLE SELECTORS OF MULTIMEA SURES
RICHARD ALO’
Department of Mathematics
Lamar University
77710 U.S.A.
Beaumont, Texas
ANDRE de KORVIN, and CHARLES E. ROBERTS, JR.
Department of Mathematics
Indiana State University
Terre Haute, Indiana 47809 U.S.A.
(Received October 17, 1978)
ABSTRACT.
The present work is a study of multivalued functions and measures
which have many
applications to mathematical economics and control problems.
Both have received considerable attention in recent years.
We study the set
of selectors of the space of all p- integrable, X- valued additive set functions.
This space contains an isometrically imbedded copy of
LP(x).
KEY WORDS AND PHRASS. p-integrable slectors, Mtimeasres, Banach space,
Radon-Nikodym property, Finite p-variation, L p spaces.
AMS (MOS) SUBJECT CLASSIFICATION (1970) Codes.
28A45, 28A48.
R. ALO’, A. KORVIN AND C. E. ROBERTS
210
i.
INTRODUCTION.
p
In his studies of classical function spaces such as L spaces, S. Leader
[7]
found it convenient to consider the Lp spaces of finitely additive set
functions which he denoted by Vp.
All the set functions are real valued and
L p can be isometrically embedded in V
p.
First the set functions
In [9] the work of [7] is generalized in two ways.
take values in a Banach space X.
Instead of studying p-summable functions,
This space is
Orlicz spaces of finitely additive set functions are considered.
denoted by
V(X).
Ixl p,
Of course, if we let p(x)
then
V(X)
space of all p-integrable, X valued additive set functions.
becomes the
It is shown that
L p(X) is isometrically embedded in Vp(X)
In recent years the study of multivalued functions and measures has
received considerable attention as have their applications to mathematical
economics and control problems.
We refer the reader to [I], [2], [3], [4], [6],
[8] for a small sample of the work done along these lines.
A multSmeasure is defined to be a function from a o-algebra into the set
of non-empty, bounded closed convex subsets of a Banach space X.
additivity is defined through the use of the Hausdorff metric.
.
multimeasure.
study of
for all A.
The set of selectors of
Countable
Let
denote a
plays a very important role in the
By a elector, we mean an X-valued measure m such that m(A) c
For example, Godet-Thobie (see [6]) shows that if
compact for all A, then
selectors {mn} such that
(A)
(A)
is weakly
has at least one selector and in fact a sequence of
{mn(A)}
is dense in
(A)
for all A.
Conditions are
also given on the space X that insure the existence of the sequence
{ran}.
The main purpose of the present paper is to study the set of selectors of
m that belong to
vP(x).
Denote that set by
MPm"
We assume here that p > i.
INTEGRABLE SELECTORS OF MULTIMEASURES
Now
MPm
is an important set to consider because it will be shown that it is
a convex closed subset of
i 2"
only if
,
vP(x)
that characterizes
that is
,
n}
then a sequence {m
m(A) for all A.
MP2
if and
With certain additional assump-
tions and using a Radon-Nikodym theorem of Tolstonogov
#
MPI
The first result deals with the case of a a-finite space where
the values of m are weakly compact subsets of X.
MP^m
211
of selectors in
vP(x)
B],
we show that if
exists with
{mn(A)} dense
in
This remains true if the values of m are assumed to be strongly
convex and closed instead of weakly compact.
Next the case where the underlying space is finite is considered.
certain assumptions using a Radon-Nikodym theorem of
is arrived at.
[6].
Under
Cost [3], the same result
These two results are thus along the line of results shown in
AMP which
Next a new space is naturally introduced, the space
space of all measures of the
miAi) rAi
form2 .r(Ai
where
m’
i
is the
are selectors of
Aie
and where
denotes a partition (finite), r is the underlying measure and
is the contraction of r to Ai.
AMPUl
and
AMP2 where
A result establishes the relation between
I + 2"
Also it is shown that
MPCclAMP.
rAi
AMP,
Finally
the case where m has absolutely closed convex values is considered.
2.
NOTATIONS AND PERTINENT RESULTS
The purpose of this section is to present some concepts and state some
results that are needed to show our results.
Let X denote a Banach space.
It is said that X has the Radon-Nikodym
property if very X-valued measure with finite variation has a density relative
to its variation. It is known that reflexive spaces and separable dual spaces
have that property.
Throughout this paper we assume p > I.
R. ALO’
212
A. KORVlN AND C
E
Let 7 denote a -algebra of subsets of a set
Let r denote a finitely
Let F be a finitely additive set
additive positive set function defined on E.
function from 7. into X with the property r(A)
0 implies F(A)
0, and
IF(En)
I[p(En)p-I
!Ene
sup
Ip(F,B)
.
ROBERTS
denotes a finite collection of disjoint subsets of B where B e 7.
where
and r(En) <
.
Now F is said to have finite p-variation if
’F(En)
Let F (-)
(mn)
Ip(F)
Ip(F,)
<
.
(.)], and AP(x) denote the set of finitely
r[E n
E e
n
additive set functions F mapping I into X with finite p-variation and with the
property r(A)
Let
vP(x)
0 implies F(A)
0.
be the submanifold of
functions, that is lim
(E)+0
IIF(E) II
AP(x)
consisting of r-continuous set
0.
It is shown in [9] that when p > I, Ap(X)
Vp(X).
I/p
We denote by
space under the
Np(F) Ip(F)
norm Np. Moreover
dense subset of
vP(x)
vP(x)
the set functions of the form
is a Banach
F
form a
provided that X has the Radon-Nikodym property and p > I.
Let F be a function from
of X.
It is also shown that
into the non-empty closed bounded convex subsets
If F has weakly compact values, we define, following Tolstonogov [8],
fFdr
{ffdrlfeL’(r)
and f(w)eF(w) a.e.}
Otherwise,
Fdr
cl{IfdrlfeL’(r)
and f(w)eF(w) a.e.}
Let X c denote the non-empty bounded closed convex subsets of X.
[6] we define
A + B
cI{A+B} for
AX%, B.
Following
INTEGRABLE SELECTORS OF MULTIMEASURES
If
:
E
+
Xc, then
(U)
213
called a multi-measure if
m is
-=n
"7.(Ai)
lira
[(AI)
$ (A2) $
$ (An)]
For every sequence of disjoint sets Ai e 7 where the limit is with respect to the
Hausdorff metric (It is shown in [6] that $ is an abelian and associative
operation on
Xc).
The reader is referred to [3] and [8] for Radon-Nikodym theorems pertaining
to multimeasures.
If
AXc
[A
will denote the Hausdorff distance between {0}
and A, that is
xeA
Also acoS will refer to the absolute convex envelope of a subset S of X,
that is acoS is the set of all finite sums of the form
Zi s i
with
sieS
and
7.1eil.<l
Of course, acoS denotes the closure of acoS.
property P if its dual space
3.
has a sequence
{Xn*}
that separates points of X.
MAIN RESULTS
I
/
c
denote a multi-measure from 7. into
Let
m:
X’
Finally X is said to have
X such that m(A)e(A) for all Ae7. with
all "selectors of
PROOF.
Now for
"
that are in
Let Aer. with r(A) <
vP(x). MP^m
.
Let
and let
meVP(x)
MP^m
denote all measures
that is
MP is
is a closed subset of
mieMPf
with
Np(m-mi)/0
and
the set of
vP(x).
mevP(x).
> 0 and i large enough:
lm(A)-mi(A) IIp
r(A)P -I
Since
(A)
is closed it follows that
a ’-finite space.
Let
X
<
m(A)e(A).
Thus
meMP.
Let (,Z,) denote
We
denote all weakly compact convex subsets of X.
assume that the convex hull of the average range of m is relati_velx weakly comRaqt,
that is
[ -%(K) IKe. KCE O<I(K)<J1
is relatively compact for every E
7.
214
THEOREM 2.
A
KORVIN AND C
E
ROBERTS
Let m be a multi-measure from E into Xw.
Assume that
(i)
The convex hull of the average range of m is relatively compact.
(2)
The space
(3)
For every e>O there exists 8>0 such that %(A) implies
(4)
X has the Radon-Nikodym property.
Then if Mp.
(A)
ALO’
R
X’
is separable (in particular
X’
satisfies P).
qb for p > 1, then there exists {mn} a sequence in Mp.
#
Im(A)
such that
=cl{mn(A)} for every A e I.
From (2) and [6] it follows that there exists a sequence {o
n
PROOF.
o n are selectors of
Now %(A)
0 implies
with
(A)
v
cl{On(A)}
re(A)
{0} by (3)
Thus o n <<%
/Afnd%
n(A)
for A e
for some
Y.,
o
are X-valued measures.
n
Now (4) implies
fneL/x(%
and all AsE.
By a theorem of [8], (i) implies that there exists a function F from
such that
implies
Let
(A)
fAFd%.
Thus
fAfnd%eYAFd%
for all
where
AeY..
into
We now show that this
fn(w)eF(w) a.e.
/Ahd%e/AFd%
for all
AsY..
xeF(w) if and only if
Let
<X,Xn>
{Xn*}
<
be a dense sequence in
sup<y,Xn*>
y’F (w)
This is because F(w) is convex, closed and bounded
of positive measure then there exists
>
<h(w),Xn*>
AeF. and
sup
If h(w)
F(w) on a
(A) > 0 and
over A
ye F (w)
Consequently
</Ahd%, Xn*>
.>
sup
g (w)e F (w)
fA<h,xn*>d% > /A
fA<g(w),xn*> d%
Thus
YAgd%cl{fAgd%Ig(w)sF(w)}.
Thus
fn(w)eF(w)
a.e. for all n.
>.
sup
yeF (w)
<Y,Xn*>d%
< Y Agd%,
sup
g (w)e F (w)
This contradicts
Then
for all n.
n such that %
<Y,Xn*>
X’.
Xn*>
/Ahd%S/AFd%
for all
set
215
INTEGRABLE SELECTORS OF MULTIMEASURES
Now let
Let
moeMPm then mo(A)
partition of
.
mj,],keVP(x)
also
mj,],k(A)
mj l,k e
MP^m since
oj(AAkf-!Aj,])
All finite sums of the form Y.
k
(A) (for
j and
%(A’
+ X B’j,],kfo
XBj,],kfj
sj,],k
Define
mj,],k(A)
Let {Ak} be a
Let
Aj,]Ak
Bj,],k
sj,],keL
is a positive integer.
%(Ak)<=.
such that
foeLPX-
and by [9],
{wl]-l<llfjIl.<]} where
Aj,]
clearly
fAfod%
fA sj,]’kd%
+
By [9]
mo(AnB’j,],k )e(A).
(A.Ak[ Aj,])
+ E(A.
k
Ak(.Aj,])
are dense in
fixed) and since
/
0 as
/
(for j fixed) it is easy to see that (3) implies
that all finite sums of the form
E
k
m,],k(A)j
are dense in
(A)
range over the
as j and
positive integers.
These finite sums form the desired sequence {m }.
n
This completes the proof.
Let X s denote all bounded closed non-empty, strictly convex subsets of X,
that is
Ae
implies that for all
x’ (x)
COROLLARY 3.
x’eX’ there
exists a unique xeX such that
sup {x’ (y)
Let (,E,%) be a o-finite measure space.
Let
be a
multimeasure from E into X s assume
is of the form
(A)
fAFd% where F(w)ec.
(2)
X’
(3)
X has the Radon-Nikodym property.
Then if
is separable.
MP #
n}
for p > i, then there exists a sequence {m
in
MP such that
R. ALO’, A. KORVIN AND C. E. ROBERTS
216
(A)
cl{mn(A)}
for all A e E.
From a result in [6] and since m has values in X
PROOF.
{n
the sequence
the existence of
s
as defined in the proof of theorem 2 follows.
The rest of the
proof follows as in Theorem 2.
We now consider a measure
Il
sup
l(Ai)
from E into X
c
We define the variation of
by
where the sup is over finite collections of disjoint
sets Ai.
We consider the case where (,E,) is a finite measure space.
THEOREM 4.
from E into X
c
(I)
Let (,E,) be a finite measure space.
has u-finite variation.
(A)
{0}.
(a)
(3)
X’
(4)
X has the Radon-Nikodym property.
Then if
PROOF.
0 implies
is separable.
MP #
(A)
for p > 1 then there exists a sequence {mn} in
cl{m (A)} for all A e E.
n
From a result in [2] and since
takes values in
is uniformly continuous with respect to
.
Since
result of [3] it follows that there exists a function F from
(A)
be a multimeasure
Assume:
(2)
holds,
Let
fAFd
for all A
c
MP such that
and since (2)
is finite by a
into Xc such that
E.
.
The rest of the proof proceeds as in Theorem 2.
We now show that
COROLLARY 4.
Let
MP
i
characterizes
and
2
be two multimeasures.
theorem 2 or theorem 4
MPI MP2
if and only if
I 2"
Under the hypothesis of
INTEGRABLE SELECTORS OF MULTIMEASURES
PROOF.
217
The proof follows immediately from the statement
PROPOSITION 5.
Let
and having no atoms.
be a multimeasure from 7. into
c
(A)
cl{mn(A)}.
of finite variation
Assume X has the Radon-Nikodym property then
MP
is a
convex set.
PROOF.
all A e 7..
From a result in [2] and since m has no atoms,
It is then straight-forward to check that
MP
(A)
is convex for
is convex.
From now on y will denote % when the hypothesis of theorem 2 are verified
and
when the hypothesis of theorem 4 are verified.
Recall that we use the
word "partition" as in [9] that is a finite collection of disjoint subsets of
finite
measure.
THEOREM 6.
MP^m
Assume the hypothesis of theorem 2 or of theorem 4 and assume
Then there exists a sequence {m i} in Mp
#
meMP there exists a partition 0
subset
{min}
of
{mj}
such that if
YEn denotes
PROOF.
refines
refines
then there is a
such that
Im-ymi
where
such that for e > 0 and
(En)
the contraction of Y to En.
Let e > 0,
meMP.
m
nl
By [9] there exists a partition
0
such that if
0 then
By theorem 2 or theorem 4 there exists a sequence {m i} in
m(E)
cl{mi(E)}
for all E
Let e’ > 0 then there exists an integer i
(En)-m(En)ll
<
n
E.
such that
e’.
MP^m
such that
_
218
Now
R. ALO’
X m(En
Np
Ene
z min(En
y (En)
YEn
(En)
A. KORVIN AND C
Ene
YEn
Y(En
]
Ip m(En)-min(En)
Y(En)
<
clearly
e’P
Y(En)P
e’ may
YEn
Aje
sup
^
YE
-
ROBERTS
n
[m(En
/.
sup
E
-min (mn)
p
YEn(AJ) p
Y(E )PY(Aj) -i
Aje
j
(--- p-I YEn(A]) -< e ’p
lYE
[ y(Aj)
Y(En)P_ I
]
be picked so that
(En)
Np /. m(En)-min
YEn
Y(En)
Ene
This finishes the proof.
AMP
YAi where m’ ieMP"
Motivated by the above theorem we define
sums of the form
E
Aie
If
I
and
2
are multimeasures from E into
(i + 2) (A)
Also let
\’
Y(Ai)
That is the partition
Let
YAi
with
is kept the same for
I + 2"
cI{AMPI Js AMP2 }.
THEOREM 7.
we define
the set of all finite sums of the form
mii(Ai) + m’2i (Ai)
Ai
c
I(A) $ 2(A).
AMPUl s AMP2 be
z_
cI{AMP}
i(Ai)
Y (A i)
to be the set of all finite
mli
I
eMPI, m2 ieMP2
and
2"
Under the hypothesis of theorem 2 or theorem 4
219
INTEGRABLE SELECTORS OF MULTIEASURES
By theorem 2 or theorem 4 there exists sequences {mi}
PROOF.
{m2i}
MP,
m,
(A)
MPI MP2
such that
cl{mli(A)} 2(A)
cl{mi(A)} I(A)
Consider y. m
i(Ai
YAi,
Y(Ai)
{mli}
for A
cl{m2i(A)}
then as in theorem 6 one may pick
e’
> 0 small enough
so that
lm i(Ai)-mi(Ai)ll<e
implies
m, i(Ai
Np
cl{mli(A)+m2j (A)} again let
Imi(Ai)-(mlki+m21i)(Ai)l l<e "where e" is
mi(Ai)
Y"
Np
Ai
clAMP& el [AMPul
’
Let us note that if
AMP2]
i
hypothesis,
Let
2
and
MPm
, MP2
{Ai}
let
#
< e.
be a multimeasure from
AieE_m’i(Ai)y(Ai) YAi
into
c,
The following are true.
(i)
If
ic valued multimeasure,
2
M
define (a--d
where
)(A)
then so is aco
p c IAMP.
Then
(A)e(A)
Y (AinA)
Aie -i
We would like to show
(Ai)
i= i
sequence of disjoint sets A i.
whenever
.<
m’leMP
a--d-((A)).
.
{Ai} be a partition and assume that the values of
absolutely convex sets in
PROOF.
satisfies the above
$ then whenever
PROPOSITION 8.
Let
This shows that
do.
be the partition
is an
YAi]
m’ li (Ai)+m’ 2j (Ai)el (Ai)$2 (Ai)’
Since
#
1
small enough so that
(mlki+m21i)(Ai)
-Y(Ai)
E
the converse is also clear.
(3)
YAi!
(A)
Since
Now let
mi (Ai)
YAi
(Ai
are
is a partition
i
E
i= i
ac--- (Ai)
for any
of A with
220
R
ALO’
A
KORVIN AND C
E
ROBERTS
Now the proof in [5] (p. 415) shows with obvious modifications that
aco(A+B)
aco A
+
aco B.
co(A+B).
aco A
+
aco B and by continuity of
ado A
+
Thus aco A
aco
acoA+acoB
d6 B
aco
aco B
a6o (A+B)
i--i
hd i(Ai)
By a result of [4], J[d A, dB B] .<
denotes the Hausdorff metric
aco
"E(Ai
"E
i=l
-i=l
+
(A+B) and by induction
n
[Ii=l(Ai)
Thus
J[A,B]
where A and B are in
c
and
Thus letting n go to infinity
(Ai)
This shows the first part of the theorem.
.
MP.m C clAMP
c
eAMp
follows from [9] since
F
are dense in
vP(x).
Finally let
Then
1 (A)=.AieZm:l (Ai) --Y(Ai)
(Ai).YAi/
mi (A)
If Z
y(AAi)
Y(Ai). <i
since m
y_(A,( Ai)
i(Ai)e(Ai)C (A)
(this because 0e(B) if
absolutely convex for all BeZ) this implies
(B)
is
c(A)e(A).
REFERENCES
i.
Aumann, R. J. Integrals of set-valued functions, J. Math. Anal. App,. i_2
(1965) 1-12
2
Coste’
3
Coste’
4.
Debreu, G. Integration of correspondences, Proc. Fifth Berke.ley Sympos. on
Mth. Statist. and Probability, Vol. 2, Part 1 (1966) 351-372.
5.
Dunford, N. and J. T. Schwartz, Linear Operators, Part i, New York, 1958.
6.
Godet-Thobie, C. Sur les selections de mesures generalisees, C. R. Acad. Sci.
271 (1970) AI53-AI56.
A. Sur les multimeasures a valeurs fermees bornees d’un espace
Banach, C. R. Acad. Sci. 280 (1975)A567--A570.
A. La propriete’ de Radon-Nikodym en integration multivoque, C
Acad. Sci. 271 (1970) AI515-AI518.
R
INTEGRABLE SELECTORS OF MULTIMEASURES
7.
Leader, S. The theory of Lp spaces for finitely additive set functions,
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8.
Tolstonogov, A. A. On he Rado-Nkodym and A. A. LJapunov theorems for
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Uhl, J. J. Orlicz spaces of finitely additive set functions, Studia Math.
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221