Strict topologies on measure spaces

Abstract

Let be a measurable space, let be a family of measurable subsets of it, and let  be a subspace of complex measures on that is also closed under restrictions of measures. In this paper we introduce the -convergence topology (, ) and thestrict topology (, ) on . Among other results, we find necessary and sufficient conditions for Hausdorff-ness and coincide-ness of these topologies. Applications to Lebesgue spaces, and also examples in Hausdorff topological spaces and locally compact groups are given. K E Y W O R D SLocally convex space, strict topology, measure space, radon measure M S C ( 2 0 1 0 ) 22D05, 28A33, 46A03, 46E27, 54A10, 54A20 Mathematische Nachrichten. 2017;290:3020-3028.