Strict and Mackey topologies and tight vector measures

Abstract

Let B() denote the Banach algebra of all bounded Borel measurable complex functions defined on a topological Hausdorff space X, and Bo() stand for the ideal of B() consisting of all functions vanishing at infinity. Then B() is a faithful Banach left Bo()-module and the strict topology β on B() induced by Bo() 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 Tm : B() → E. It is shown that a measure m : → E is countably additive tight if and only if the corresponding integration operator Tm 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.