Strict and Mackey topologies and tight vector measures

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 beta on B() induced by B-o() is a mixed topology. For a sequentially complete locally convex Hausdorff space (E, xi), 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 (eta, xi)-continuous, where eta denotes the infimum of the strict topology beta and the Mackey topology tau (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 tau (B(), ca())-neighborhood U of 0 and some beta-neighborhood W of 0 in B(). As an application, we prove a Nikodym type convergence theorem for countably additive tight vector measures.