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 ${\sum }_{n=1}^{\infty }\Vert v_n\Vert {\mu }_n(X)<\infty$, 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.