Let be a topological Hausdorff space and be the -algebra of Borel sets in . Let be the space of all bounded -measurable scalar functions on , equipped with the Mackey topology , where denotes the Banach space of all scalar Radon measures on . It is proved that is a strongly Mackey space. For a sequentially complete locally convex Hausdorff space , let denote the space of all Radon measures , equipped with the topology of setwise convergence. It is proved that a subset of is relatively -compact if and only if is uniformly regular and for each , the set in is relatively -compact, if and only if the family of corresponding integration operators is -equicontinuous and for each , the set in is relatively -compact. As an application, we get a Nikodym theorem and a Dieudonné–Grothendieck type theorem on the setwise sequential convergence in .
Cite this article
Marian Nowak, Strongly Mackey topologies and Radon vector measures. Port. Math. 77 (2020), no. 3/4, pp. 283–297DOI 10.4171/PM/2052