Prokhorov theorem in locally compact Hausdorff space? 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!
 A: There is a version of Prokhorov's theorem on locally compact Hausdorff spaces, but it seems surprisingly difficult to locate it in the literature.
One reference (that simultaneously gives both the LCH and Polish space versions) is Bourbaki's Intégration. Chapitre IX, Section 5.5, Théorème 1 + Théorème 2 (each theorem gives one implication). I'll summarize.
Definition (Prokhorov condition) Let $H$ be a collection of complex measures on a topological space $X$. $H$ is Prokhorov if

*

*the total variations $|\mu|(X)$ of $\mu\in H$ are uniformly bounded, and


*for every $\varepsilon >0$ there is a compact $K=K_{\varepsilon}\subseteq X$ such that $|\mu|(X-K)<\varepsilon$ for all $\mu\in H$.
Then:
Theorem 1 (Prokhorov $\Rightarrow$ compact) A Prokhorov set of complex measures on a completely regular space $X$ (in particular an LCH space or a metric space) is relatively compact in the weak$^*$ topology on complex measures relative to the bounded continuous functions on $X$. 
Theorem 2 (compact $\Rightarrow$ Prokhorov) If $X$ is LCH or complete metric separable then a weak$^*$-compact set of positive finite measures on $X$ is Prokhorov.
