p INTEGRABLE SELECTORS OF MULTIMEA SURES

. 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).

L p can be isometrically embedded in V p.
In [9] the work of [7] is generalized in two ways. First the set functions take values in a Banach space X. Instead of studying p-summable functions, Orlicz spaces of finitely additive set functions are considered. This space is denoted by V(X). Of course, if we let p(x) Ixl p, then V(X) becomes the space of all p-integrable, X valued additive set functions. It is shown that Lp(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. Countable additivity is defined through the use of the Hausdorff metric. Let denote a multimeasure. The set of selectors of plays a very important role in the study of . By a elector, we mean an X-valued measure m such that m(A) c (A) for all A. For example, Godet-Thobie (see [6]) shows that if (A) is weakly compact for all A, then has at least one selector and in fact a sequence of selectors {m n} such that {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 MPm" m that belong to vP(x). Denote that set by We assume here that p > i. Now MP is an important set to consider because it will be shown that it is m a convex closed subset of vP(x) that characterizes , that is MPI MP2 if and only if i 2" The first result deals with the case of a a-finite space where the values of m are weakly compact subsets of X. With certain additional assump- Next the case where the underlying space is finite is considered. Under certain assumptions using a Radon-Nikodym theorem of Cost [3], the same result is arrived at. These two results are thus along the line of results shown in [6]. Next a new space is naturally introduced, the space AMP which is the miAi) AMPUl and AMP2 where I + 2" Also it is shown that MPCclAMP. Finally the case where m has absolutely closed convex values is considered.

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' A. KORVlN

I/p
We denote by Np(F) Ip(F) It is also shown that vP(x) is a Banach space under the norm Np. Moreover the set functions of the form F form a dense subset of vP(x) provided that X has the Radon-Nikodym property and p > I. Let    (2) The space X' is separable (in particular X' satisfies P). By a theorem of [8], (i) [9] mj,],keVP(x) also mj l,k e MP^m since (2) X' is separable.
(4) X has the Radon-Nikodym property.  (2) holds, is uniformly continuous with respect to . Since is finite by a result of [3] it follows that there exists a function F from into X c such that (A) fAFd for all A E.
The rest of the proof proceeds as in Theorem 2.
We now show that MP characterizes . PROOF. From a result in [2] and since m has no atoms, (A) is convex for all A e 7.. It is then straight-forward to check that MP 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. PROOF. Let e > 0, meMP. By [9]

R ALO' A KORVIN AND C E ROBERTS
Now the proof in [5] (p.