On the Riesz Integral Representation of Additive Set-Valued Maps (II)

Let T be a compact topological space, and let $C_+(T)$ be the space of all non-negative continuous real-valued functions defined on T endowed with the topology of uniform convergence. We prove the Riesz integral representation for continuous additive and positive set-valued maps defined on $C_+(T)$ with values in the space cc(E) of all weakly compact convex non-empty subsets of a Banach space E. As an application we give a generalization of Dunford-Schwartz's result on the Riesz integral representation for any continuous set-valued map (not necessary positive).