Radon-Nikodym Theorems for vector-valued finitely additive measures

This paper studies the existence of a density function in the Dunford sense for a finitely additive measure with values in a locally convex topological vector space (LCTVS) X . A finitely additive measure m from a σ-algebra to a locally convex is said to be dominated if there exists a finitely additive extended real-valued measure λ such that m<<λ. When m is dominated with control λ, a Dunford-type derivative exists if there is a function f:Ω→X′′ such that <f(⋅),y> is measurable and λ-integrable for every y in X′, the function f is bounded on each member of an increasing sequence of measurable sets that fill Ω in measure, and ∫E<f(⋅),y>dy=<y,m(E)> for every measurable set E. Here the duals of X have been endowed with the strong topology. We find several conditions involving the separability of X′, subsets of {m(A)/λ(A):A measurable}, and the weak derivatives d<y,m>/dλ that ensure the existence of Dunford-type derivatives. In the final section we give conditions for the existence of Dunford-type derivatives for Banach-space-valued finitely additive measures.