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The paper presents some weak compactness criterion for a subset $M$ of theset $\mathfrak{RM}_b(T,\mathcal{G})$ of all positive bounded Radon measures ona Hausdorff topological space $(T,\mathcal{G})$ similar to the Prokhorovcriterion for a complete separable metric space. Since for a generaltopological space the classical space $C_b(T,\mathcal{G})$ of all boundedcontinuous functions on $T$ can be trivial and so does not separate points andclosed sets, instead of $C_b(T,\mathcal{G})$-weak compactness we consider$S(T,\mathcal{G})$-weak compactness with respect to the new uniformly closedlinear space $S(T,\mathcal{G})$ of all (symmetrizable) metasemicontinuousfunctions.

Prokhorov-like conditions for weak compactness of sets of bounded Radon measures on different topological spaces