Let $X$ be a topological space and $\mathcal M(X)$ the space of all finite signed Borel measures on $X$. For $\nu \in \mathcal M(X)$, let $(\nu^+, \nu^-)$ be its Jordan decomposition, and $|\nu| := \nu^+ + \nu^-$ its associated variation. We endow $\mathcal M(X)$ with the total variation norm $[\cdot]$ defined by $[\nu] := |\nu|(X)$. Then $(\mathcal M(X), [\cdot])$ is a Banach space.
- A subset $M$ of $\mathcal M(X)$ is said to tight if for every $\varepsilon>0$ there is a compact subset $K$ of $X$ such that $\sup_{\nu \in M}| \nu| (X \setminus K) <\varepsilon$.
- A subset $M$ of $\mathcal M(X)$ is said to bounded if $\sup_{\nu \in M} [\nu] <\infty$.
Then I come across this version of Prokhorov's theorem from this French Wikipedia page.
Theorem: Let $X$ be completely regular and $M$ a subset of $\mathcal M (X)$.
- If $M$ tight and bounded, then the closure of $M$ in $\sigma(\mathcal M (X), \mathcal C_b(X))$ is sequentially compact.
- If $X$ is locally compact (or Polish) and the closure of $M$ in $\sigma(\mathcal M (X), \mathcal C_b(X))$ sequentially compact, then $M$ is tight and bounded.
This is very general. In Onno van Gaans's lecture note Probability measures on metric spaces, for (1.) to hold, it is assumed further that $X$ is separable.
Could you confirm if above theorem with such a great generality is indeed true? If yes, could you please provide a reference.
