# Strongly Mackey topologies and Radon vector measures

### Marian Nowak

University of Zielona Góra, Poland

## Abstract

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

## Cite this article

Marian Nowak, Strongly Mackey topologies and Radon vector measures. Port. Math. 77 (2020), no. 3/4, pp. 283–297

DOI 10.4171/PM/2052