Selections of integrable multifunctions in arbitrary Banach spaces

Abstract

We find the origin of the integration theory for multifunctions in the sixties in the pioneering works of G. Debreu and R. Aumann, Nobel prizes for the Economy in 1983 and in 2005, respectively. The Aumann integral is defined by means the integrals of measurable selections of the multifunctions. An important tool for the existence of measurable selections is the Kuratowski and Ryll-Nardzewski theorem, although this famous result needs the separability of the range space. Other definitions of multifunction integrals, that are not based on selections have been developed such as Pettis, Henstock, McShane, Birkhoff, Kurzweill-Henstock-Pettis, or variationally integrals. However to obtain good properties of such integrals for multifunctions one also needs to study "nice" properties of selections. This chapter is devoted to selection results in the framework of an arbitrary Banach space (non necessarily separable). In particular we address our attention to the existence of selections integrable of the same sense of the given multifunction and to their applications to the problem of representing a multifunction as a translation of a multifunction with better integrable properties by means of one of its integrable selections.