Strongly Mackey topologies and Radon vector measures

Abstract

Let $X$ be a topological Hausdorff space and $\mathcal{B}o$ be the $\sigma$-algebra of Borel sets in $X$. Let $B(\mathcal{B}o)$ be the space of all bounded $\mathcal{B}o$-measurable scalar functions on $X$, equipped with the Mackey topology $\tau (B(\mathcal{B}o),M(X))$, where $M(X)$ denotes the Banach space of all scalar Radon measures on $X$. It is proved that $(B(\mathcal{B}o),\tau (B(\mathcal{B}o),M(X)))$ is a strongly Mackey space. For a sequentially complete locally convex Hausdorff space $(E,\xi )$, let $M(X,E)$ denote the space of all Radon measures $m:\mathcal{B}o\to E$, equipped with the topology ${\mathcal{T}}_s$ of setwise convergence. It is proved that a subset $\mathcal{R}$ of $M(X,E)$ is relatively ${\mathcal{T}}_s$-compact if and only if $\mathcal{R}$ is uniformly regular and for each $A\in \mathcal{B}o$, the set $\{m(A):m\in \mathcal{R}\}$ in $E$ is relatively $\xi$-compact, if and only if the family $\{T_m:m\in \mathcal{R}\}$ of corresponding integration operators $T_m:B(\mathcal{B}o)\to E$ is $(\tau (B(\mathcal{B}o),M(X)),\xi )$-equicontinuous and for each $v\in B(\mathcal{B}o)$, the set $\{{\int }_{X}vdm:m\in \mathcal{R}\}$ in $E$ is relatively $\xi$-compact. As an application, we get a Nikodym theorem and a Dieudonné–Grothendieck type theorem on the setwise sequential convergence in $M(X,E)$.