On the range of a vector measure

Abstract

Let $(\Omega ,\Sigma ,\mu )$ be a finite measure space, $Z$ be a Banach space and $\nu :\Sigma \to Z^{\ast }$ be a countably additive $\mu$-continuous vector measure. Let $X\subseteq Z^{\ast }$ 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 $⟨X,Z⟩$. Suppose that, either $(Z,\sigma (Z,X))$ has the Mazur property, or $(B_{X^{\ast }},w^{\ast })$ 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 $w^{\ast }$-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 $Z=X^{\ast }$.