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.
تحميل البحث