TITLE: Prokhorov's theorem in non separable metric spaces
QUESTION [13 upvotes]: 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, 
Prokhorov Theorem
.
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?

REPLY [13 votes]: It is correct, see Theorem 8.6.7 in volume 2 of Bogachev's "Measure Theory" monograph. See also his Theorem 8.6.8 for a version of the second statement which covers the case of a non-separable space.