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

## TL;DR

This paper characterizes the Radon-Nikodym property for Banach spaces by linking it to the representation of vector measures of bounded variation as sums involving scalar measures and vectors.

## Contribution

It establishes a necessary and sufficient condition for a Banach space to have the Radon-Nikodym property based on vector measure representations.

## Key findings

- Banach space has Radon-Nikodym property iff all bounded variation vector measures are representable as sums of scalar measures and vectors.
- Provides conditions under which operators on positive measures extend uniquely to vector measures in spaces with the Radon-Nikodym property.
- Connects the Radon-Nikodym property to the structure of vector measures and operator extensions.

## 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.