Strict topologies on measure spaces

Let X be a measurable space, let be a family of measurable subsets of it, and let be a subspace of complex measures on X that is also closed under restrictions of measures. In this paper we introduce the ‐convergence topology and the ‐strict 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.