A nonuniform version of the theorem of Radon-Nikodým in the finitely additive case with applications to extensions of finitely additive set functions

Abstract:

For $\mu ,\nu \in b{a_ + }(\Omega ,\mathfrak {A})$ it is shown that the existence of a net of nonnegative functions ${f_{\alpha ’}}$ that are primitive relative to $\mathfrak {A}$ and satisfy ${\lim _\alpha }{\smallint _A}{f_\alpha }d\mu = \nu (A),A \in \mathfrak {A}$, is equivalent to the condition $\nu \lesssim \mu$, i.e. $\mu (A) = 0$ for some $A \in \mathfrak {A}$ implies $\nu (A) = 0$. Furthermore, as an application it is proved that for $\mu ,\nu \in b{a_ + }(\Omega ,\mathfrak {A})$ satisfying $\nu \lesssim \mu$ and any extension $\mu ’ \in b{a_ + }(\Omega ,\mathfrak {A}’)$ of $\mu$, where $\mathfrak {A}’$ denotes some algebra of subsets of $\Omega$ containing $\mathfrak {A}$, there exists some extension $\nu ’ \in b{a_ + }(\Omega ,\mathfrak {A}’)$ of $\nu$ such that $\nu ’ \lesssim \mu ’$ is valid.