Characterization of the closed convex hull of the range of a vector-valued measure

AbstractFor a set K in a locally convex topological vector space X there exists a set T, a σ-algebra S of subsets of T and a σ-additive measure m: S → X such that K is the closed convex hull of the range {m(E): E ∈ S} of the measure m if and only if there exists a conical measure u on X so that K Ku,Ku, the set of resultants of all conical measures v on X such that v < u