A characterization of the Radon-Nikodym property for vector valued measures

Abstract

If $\mu_1,\mu_2,\dots$ are positive measures on a measurable space $(X,\Sigma)$ and $v_1,v_2, \dots$ are elements of a Banach space ${\mathbb E}$ such that $\sum_{n=1}^\infty \|v_n\| \mu_n(X) < \infty$, then $\omega (S)= \sum_{n=1}^\infty v_n \mu_n(S)$ defines a vector measure of bounded variation on $(X,\Sigma)$. We show ${\mathbb E}$ has the Radon-Nikodym property if and only if every ${\mathbb E}$-valued measure of bounded variation on $(X,\Sigma)$ is of this form. As an application of this result we show that under natural conditions an operator defined on positive measures, has a unique extension to an operator defined on ${\mathbb E}$-valued measures for any Banach space ${\mathbb E}$ that has the Radon-Nikodym property.