A Radon-Nikodym theorem for a pair of Banach-valued finitely additive measures

One of the most interesting problems arising in the study of finitely additive measures (f.a.m.) concerns the existence of a Radon-Nikodým derivative for a pair of f.a.m. /λ/,/m/, with /λ/<</m/. It is known that the classical Radon-Nikodým theorem fails to be true in the finitely additive case unless some further assumption is fulfilled. The first result in this direction dates back to H. B. Maynard, who investigated the case of two scalar f.a.m. defined on an algebra of sets. The scalar case as well as the vector-valued case have been studied by various authors. In this paper, using a recent integration theory with respect to a Banach-valued finitely additive measure, we extend Maynard's result for a pair of Banach-valued f.a.m.; to do this we make use of a condition that is equivalent to that assumed by Maynard and by J. W. Hagood, but which turns out to be strictly stronger in the case here considered, as shown by means of an example.