Mackey topologies and compactness in spaces of vector measures

Abstract

Let $\Sigma$ be a $\sigma$-algebra of subsets of a non-empty set $\Omega$. Let $B(\Sigma)$ be the space of all bounded $\Sigma$-measurable scalar functions defined on $\Omega$, equipped with the natural Mackey topology $\tau(B(\Sigma),ca(\Sigma))$. Let $(E,\xi)$ be a quasicomplete locally convex Hausdorff space and let $ca(\Sigma,E)$ be the space of all $\xi$-countably additive $E$-valued measures on $\Sigma$, provided with the topology ${\cal T}_s$ of simple convergence. We characterize relative ${\cal T}_s$-compactness in $ca(\Sigma,E)$, in terms of the topological properties of the corresponding sets in the space ${\cal L}_{\tau,\xi}(B(\Sigma),E)$ of all $(\tau(B(\Sigma),ca(\Sigma)),\xi)$-continuous integration operators from $B(\Sigma)$ to $E$. A generalized Nikodym type convergence theorem is derived.