About the metrizability of the space of Probability measures $\mathcal{P}(S)$

Question

It is often proved in Books that the space of Probability measures $\mathcal{P}(S)$ on a Polish metric space $(S,\rho)$ endowed with the weak/narrow topology induced by declaring it to be be the coarsest topology on $\mathcal{P}(S)$, which makes the mappings $$\mathcal{P}(S) \ni \mu \mapsto \int f d\mu \in \mathbb{R}$$ continuous for each bounded and continuous $f : S \rightarrow \mathbb{R}$, is metrizable. Two such metrics should be the Prokhorov metric $d_P$ and the Wasserstein metric $W_0$ of the bounded distance function $\min\{\rho,1\}$.

But the thing I do not understand is the following: It is often shown (For example in Villani, 2009, Optimal Transport) that $(\mu_n) \subset \mathcal{P}(S)$ converging weakly to some $\mu \in \mathcal{P}(S)$, that is, $$\int f d\mu_n \rightarrow \int f d\mu$$ for each bounded and continuous $f : S \rightarrow \mathbb{R}$, is equivalent to $W_0(\mu_n,\mu) \rightarrow 0$, or $d_P(\mu_n,\mu)\rightarrow 0$. If we do not know a priori that the weak topology is metrizable, then we cannot conclude by the above, that the topology generated by $W_0$ or $d_P$ is exactly the weak topology. Or am I missing something?

Answer 1

I'm not really sure what Villani wrote in his monograph, but it is true that one needs to prove that the weak topology is induced by a distance, as a priori it could be another topology with the same converging sequences.

This is quite standard, though. The key point is to realize that it is sufficient to check the continuity of $\mu\mapsto\int f d\mu$ for a countable number of continuous and bounded functions. This is quite obvious if the space is compact (because in this case $C_b(S)$ is separable), but possible also for general Polish spaces.

The details can be found, e.g., in Section 8.3 of the second book by Bogachev "Measure Theory" or at the beginning of my book with Pasqualetto "Lectures on Nonsmooth Differential Geometry"