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?