On the range of a vector measure

TL;DR

This paper investigates conditions under which the range of a vector measure in a Banach space is contained in a specific subspace, extending previous results by relaxing assumptions on the space and measure properties.

Contribution

It extends existing theorems by establishing new criteria for the range of vector measures to lie within a subspace under weaker topological and compactness assumptions.

Findings

Range of vector measure contained in subspace under specified conditions

Extension of Freniche's results to broader settings

Conditions involving convex hulls and topological properties are sufficient

Abstract

Let $(\Omega,\Sigma,\mu)$ be a finite measure space, $Z$ be a Banach space and $\nu:\Sigma \to Z^*$ be a countably additive $\mu$-continuous vector measure. Let $X \subseteq Z^*$ be a norm-closed subspace which is norming for $Z$. Write $\sigma(Z,X)$ (resp. $\mu(X,Z)$) to denote the weak (resp. Mackey) topology on $Z$ (resp. $X$) associated to the dual pair $\langle X,Z\rangle$. Suppose that, either $(Z,\sigma(Z,X))$ has the Mazur property, or $(B_{X^*},w^*)$ is convex block compact and $(X,\mu(X,Z))$ is complete. We prove that the range of $\nu$ is contained in $X$ if, for each $A\in \Sigma$ with $\mu(A)>0$, the $w^*$-closed convex hull of $\{\frac{\nu(B)}{\mu(B)}: \, B\in \Sigma, \, B \subseteq A, \, \mu(B)>0\}$ intersects $X$. This extends results obtained by Freniche [Proc. Amer. Math. Soc. 107 (1989), no. 1, 119--124] when $Z=X^*$.