On finitely additive measures in nuclear spaces

The paper deals with Radon-Nikodým theorems for a finitely additive set function m taking values in a quasi complete dual nuclear space X. We start with some results concerning the structure of finitely additive measures with values in X, and next we prove the existence of a Bochner integrable density for an X –valued finitely additive m with respect to some scalar positive finitely additive measure μ whenever m<<μ and the following assumptions hold: the set of quotients {m(A)/μ(A): μ(A)>0} is bounded in X and all the scalar measures <x′|m>, x′ in the dual X′, admit bounded densities with respect to μ. It is also shown that, for a σ-field and μ σ-additive, the conclusion holds for m<<μ without further assumptions.