A Radon-Nikodym theorem for multimeasures

The starting point of this note is the article of Castaing, Touzani and Valadier: in it the authors obtain an elegant characterization of those multivalued finitely additive measures admitting approximated densities with respect to a scalar finitely additive measure. This approximated densities turn out to be in fact ”simple” multifunctions. Since every classical abstract integration theory makes use of simple functions as approximating tools, it seemed rather natural to develop an integration of this type in in a locally convex topological vector space X. In the third section we obtain an exact Radon-Nikodym Theorem for such integration, under conditions of the classical Maynard-type. Our Theorem is new even for single valued finitely additive measures theory for multifunctions F with closed convex bounded values in a locally convex topological vector space X, with respect to a finitely additive measure μ.