Contributions to the theory of set valued functions and set valued measures

Abstract

Measurable multifunctions and multimeasures with values in a Banach space are studied. We start by proving a variation of the known Dunford theorem for weak compactness in L 1 ( X ) {L^1}(X) . With a similar technique we prove that the range of certain vector valued integrals that appear in applications is w w -compact and convex. Also we obtain Dunford-Pettis type theorems for sequences of integrably bounded multifunctions. Some pointwise w w -compactness theorems are also obtained for certain families of measurable multifunctions. Then we prove a representation theorem for additive, set valued operators defined on L 1 ( X ) {L^1}(X) . Finally, in the last section, a detailed study of transition multimeasures is conducted and several representation theorems are proved.