A survey on multivalued integration

Robert J. Aumann was the first who introduced in 1965, in the paper ”Integrals of set valued functions”, the definition of a set valued integral with respect to the Lebesgue measure . In his work he considered multifunctions F and defined the integrals in terms of integrable selections. The Aumann integral, that we denote from now on by (A)-integral, obviously always exists; nevertheless it can be empty. In any case this formulation, which is very natural and useful in view of applications (as in Control Theory or Mathematical Economics), lacks in general a lot of properties. For instance: convexity, closedness, compactness and convergence theorems, which are very important in the mentioned applications. So after the pioneer work of Aumann the problem of multivalued integration was extensively studied. I will mention here only the multivalued integration introduced by Gerard Debreu. The Debreu integral is an extension of the Bochner integral to the case of set valued functions. In his important paper Debreu also compared his integral with the (A)-integral and showed that the two integrals coincide when the multifunctions take values in the hyperspace of compact convex subsets of IRn with the Hausdorff distance h. In this way what is true for the (D)-integral is true for the (A)-integral. Following this idea the multivalued integration has been extended to the case of infinite dimensional spaces either in countably additive or in finitely additive framework. The aim of this paper is to produce a short history about the ”equivalent” definition of the Debreu integral.