MACKEY TOPOLOGIES AND COMPACTNESS IN SPACES OF VECTOR MEASURES

Let Sigma be a sigma-algebra of subsets of a non-empty set Omega. Let B (Sigma) be the space of all bounded Sigma-measurable scalar functions defined on Omega, equipped with the natural Mackey topology tau(B(Sigma), ca (Sigma)). Let (E,xi) be a quasicomplete locally convex Hausdorff space and let ca (Sigma, E) be the space of all xi-countably additive E-valued measures on Sigma, provided with the topology T-s of simple convergence. We characterize relative T-s -compactness in ca (Sigma, E), in terms of the topological properties of the corresponding sets in the space L-tau,L-xi(B(Sigma); E) of all (tau(B(Sigma),ca (Sigma)), xi)-continuous integration operators from B(Sigma) to E. A generalized Nikodym type convergence theorem is derived.