Recently, working in some calculations I needed to use the Prokhorov's theorem about compactness for probability measures. However, a friend warned me that I had not the hypotesis of separability required by the theorem.
After some searching, over the books which I have reach, this is the version of the theorem that I found:
Let $ {\displaystyle (S,\rho )} $ be a separable metric space. Let $ {\displaystyle {\mathcal {P}}(S)}$ denote the collection of all probability measures defined on ${\displaystyle S}$ (with its Borel σ-algebra).
Theorem (Prokhorov). A collection ${\displaystyle K\subset {\mathcal {P}}(S)}$ of probability measures is tight if and only if the closure of ${\displaystyle K}$ is sequentially compact in the space ${\displaystyle {\mathcal {P}}(S)}$ equipped with the topology of weak convergence.
However, in a internet research I found the following document,
where no separability is required to get one of the directions of the theorem, here a description of the result contained in this notes:
Theorem. Let $S$ be a metric space. If collection ${\displaystyle K\subset {\mathcal {P}}(S)}$ of probability measures is tight then the closure of ${\displaystyle K}$ is sequentially compact in the space ${\displaystyle {\mathcal {P}}(S)}$ equipped with the topology of weak convergence. Conversely if $S$ is separable and complete, then each relatively compact set is tight.
Question: Is this version correct? If so, can someone provide me some reference?
