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Prokhorov theorem gives a compactness condition in the space of probability measures on a Polish space. I am wondering whether we have similar conditions for probability measures on more general spaces, say, locally compact Hausdorff spaces, which seems to me to be a more natural setting of measure theory.

However, since the proofs that I have seen for Prokhorov theorem depend heavily on the completeness and separability of the underlying space, they do not help much when one tries to extend the result to more general spaces. And to my best guess such an extension would rely on techniques from functional analysis.

So, do we actually have such a condition for more general spaces?

Thanks!

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  • $\begingroup$ I should pose this as a question. My impression is that Prokhorov's is a statement about sequential compactness. The set of probability measures, as the dual of $C_0(X)$, is compact in the weak-$^*$ topology for any LCH $X$ by the Banach-Alaoglu theorem. To make the weak-$^*$ topology metrizable, one needs separability conditions on $X$ or imposes restrictions on the measures considered (like uniform tightness). $\endgroup$ – Michael Nov 16 '13 at 3:12

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