Let 𝑋 be a measurable space, let 𝔅 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 𝔅-convergence topology 𝜏(, 𝔅) and the 𝔅- strict topology 𝛽(, 𝔅) 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.
