A Radon-Nikodým theorem and \(L_ p\) completeness for finitely additive vector measures (Q1080970)

Let \(\mu\) be a bounded finitely additive real-valued measure and \(\nu\) be a finitely additive measure with values in a Banach space, defined on a common field of sets. Necessary and sufficient conditions are obtained for the existence of a \(\mu\)-integrable function f such that \(\nu (E)=\int_{E}fd\mu\) for all sets E in the field. The conditions involve absolute continuity, properties of the range of \(\nu\) /\(\mu\), and certain exhaustion and boundedness properties investigated by the author. Using the main results, it is also shown that those bounded finitely additive measures for which the corresponding \(L_ p\) spaces of vector-valued functions are complete are precisely the ones for which each pair of disjoint components is separated in the sense of concentration on ideals.