Article

Strict and Mackey topologies and tight vector measures

Published in: Quaestiones Mathematicae
Volume 42 , issue 1 , 2019 , pages: 113–124

DOI: 10.2989/16073606.2018.1440443

Author(s): Marian Nowak Faculty of Mathematics, Computer Science and Econometrics, Poland

Keywords: 46G10 , 28A32 , 28A25 , 46A70 , 46G10 , 28A32 , 28A25 , 46A70

Abstract

Let B() denote the Banach algebra of all bounded Borel measurable complex functions defined on a topological Hausdorﬀ space X, and B_o() stand for the ideal of B() consisting of all functions vanishing at infinity. Then B() is a faithful Banach left B_o()-module and the strict topology β on B() induced by B_o() is a mixed topology. For a sequentially complete locally convex Hausdorﬀ space (E, ξ), we study the relationship between vector measures m : → E and the corresponding continuous integration operators T_m : B() → E. It is shown that a measure m : → E is countably additive tight if and only if the corresponding integration operator T_m is (η, ξ)-continuous, where η denotes the infimum of the strict topology β and the Mackey topology τ (B(), ca()). If, in particular, E is a Banach space, it is shown that m is countably additive tight if and only if Tm(absconv(U ∪ W)) is relatively weakly compact in E for some τ (B(), ca())-neighborhood U of 0 and some β-neighborhood W of 0 in B(). As an application, we prove a Nikodym type convergence theorem for countably additive tight vector measures.

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