On the range of a vector measure

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 $Z=X^*$.