On the range of a vector measure

Abstract

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