On the comparison of Aumann and Bochner integrals

In this paper we introduce multivalued integration in a separble Banach space X with respect to a finitely additive measure. In particular we introduce the Aumann integral with respect to the finitely additive measure and we compare it with the more usual Bochner integral; in particular we obtain the equivalence for totally measurable integrands with compact and convex values, via Stone extensions, that as usual do preserve the Bochner integral. In this way the Aumann integral turns out to be a "genuine" integral, enjoying the most usual propertie that an integral is expected to have.