Strict and Mackey topologies and tight

Abstract

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

Mathematics Subject Classication (2010): 46G10, 28A32, 28A25, 46A70.

Key words: Tight vector measures, strict topologies, mixed topologies, Mackey topologies, Banach B-modules, generalized DF-spaces.