On the range of a vector measure

Abstract:

Let $(\Omega ,\Sigma ,\mu )$ be a finite measure space, let $Z$ be a Banach space, and let $\nu :\Sigma \to Z^*$ 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, pp. 119–124] when $Z=X^*$.