On a new type of weak compactness of sets of bounded Radon measures on arbitrary Hausdorff spaces

The talk is devoted to a new kind of weak compactness of sets of bounded Radon measures appropriate for arbitrary Hausdorff spaces, where the linear space of bounded continuous functions may be trivial. As an adequate substitute of this space, the uniformly closed linear space of metasemicontinuous functions is proposed. The last space is generated by the set of bounded semicontinuous functions. A criterion closed to the famous Prokhorov criterion is obtained for this kind of weak compactness.
The proof of this criterion is based on the results describing Radon integrals as linear functionals obtained by the authors and expounded in their book «Sets. Functions. Measures» (vol. 1 and 2, Walter de Gruyter, 2018).