A Representation Theorem for Aumann Integrals

In recent years the study of set-valued functions has been developed extensively by many authors, with applications to mathematical economics and control theory; see Refs. [5, 11, 15, 161. In those papers, three approaches can be distinguished according to whether the range space (values of setvalued functions) is .Z”, a Banach space, or a locally convex topological space. The purpose of this paper is to establish properties of Aumann’s integrals of set-valued functions, F: T+ 2’, whose values are nonempty subsets of a real separable reflexive Banach space X, and to continue the work due to Aumann [2] and Datko [7-81. While previous analysis has always treated the case of special finite nonatomic measure spaces, we focus here on the case of general a-finite nonatomic measure spaces. In this last situation, moreover, the analogous results we establish hold under less stringent hypotheses. More precisely, all through the paper we consider a measure space (T, C, p), where p is supposed to be positive, nonatomic and u-finite, and we give the following statements.