The Section 5 of the book:

**Billingsley, P., Convergence of Probability Measures, 1999**,

studies Prohorov's theorem. A short reminder is given below.

Let $\Pi$ be a family of probability measures on $(S,\mathcal{F})$. We call $\Pi$ *relatively compact* if every sequence of elements of $\Pi$ contains a weakly convergent subsequence. The family $\Pi$ is *tight* if for every $\epsilon$ there is a compact set $K$ such that $P(K)>1-\epsilon$ for every $P$ in $\Pi$.

**The direct half** of the Prohorov's theorem is given in the Theorem 5.1: If $\Pi$ is tight, then it is relatively compact.

**The converse half** of Prohorov's theorem is given in the Theorem 5.2: Supose that $S$ is separable and complete. If $\Pi$ is relatively compact, then it is tight.

**My question:** In the proof of the Theorem 5.2 (i.e. relatively compact $\Rightarrow$ tight), we use separability and completness of the space $S$. On the other hand, in the proof of the Theorem 5.1 (i.e. tight $\Rightarrow$ relatively compact), I know that we do not need completness of $S$, but I do not know if we do need separability. I didn't find the place where separability is used in the proof of the Theorem 5.1. So my question is do I need or not the separability of the space $S$ in the direct part of the Prohorov's theorem?

**Remarks:**

- I know the proofs of the same theorem that use separability (e.g. Note).
- Prohorov's theorem in most books is given as one theorem on Polish spaces, so they assume separability in both halfs. It goes like this usually: Let $S$ be a Polish space and $\Pi$ a collection of probability measures on $S$. Than $\Pi$ is tight if and only if it is relatively compact.

The reason I am asking is that I would like to use the direct half of Prohorov's theorem on the problem I am currently working. Space $S$ in my case is complete but not separable.

Help with this would be great and needed. Thanks in advance.