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29 March 2024
 
  » arxiv » 1906.3658

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Prokhorov-like conditions for weak compactness of sets of bounded Radon measures on different topological spaces
Valeriy K. Zakharov ; Timofey V. Rodionov ;
Date 9 Jun 2019
AbstractThe paper presents some weak compactness criterion for a subset $M$ of the set $mathfrak{RM}_b(T,mathcal{G})$ 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 $C_b(T,mathcal{G})$ of all bounded continuous functions on $T$ can be trivial and so does not separate points and closed sets, instead of $C_b(T,mathcal{G})$-weak compactness we consider $S(T,mathcal{G})$-weak compactness with respect to the new uniformly closed linear space $S(T,mathcal{G})$ of all (symmetrizable) metasemicontinuous functions. The $S(T,mathcal{G})$-weak topology on $mathfrak{RM}_b(T,mathcal{G})$ is much weaker than the known topology $mathcal{T}_s$ of setwise convergence with respect to the $sigma$-algebra $mathcal{B}$ of all Borel subset of $T$.
Source arXiv, 1906.3658
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