Аннотация:The paper presents some weak compactness criterion for a subset~$M$ of the set $\RM_b(T,\cG)$ of all positive bounded Radon measures on a Hausdorff topological space $(T,\mathcal{G})$ similar to the Prokhorov criterion for a complete separable metric space.Since for a general topological space the classical space $C_b(T,\cG)$ of all bounded continuous functions on~$T$ can be trivial and so does not separate points and closed sets, instead of $C_b(T,\cG)$-weak compactness we consider $S(T,\cG)$-weak compactness with respect to the new uniformly closed linear space $S(T,\mathcal{G})$ of all (symmetrizable) metasemicontinuous functions.The $S(T,\cG)$-weak topology on $\RM_b(T,\cG)$ is much weaker than the known topology~$\mathcal{T}_s$ of setwise convergence with respect to the $\si$-algebra~$\cB$ of all Borel subset of~$T$.