A basic question on weak convergence of measures

Question

Why do we need separability of the space to talk about weak convergence of measures ?

Answer 1

It is not true that the definition of the concept of weak convergence of measures requires separability. We can indeed define the weak convergence on any metric space $(X,d)$, saying that the sequence of Borel probability measures $(\mu_n)_{n\geqslant 1}$ weakly converges to $\mu$ if for each $f\colon X\to \mathbb R$ continuous and bounded, $$\lim_{n\to \infty}\int_Xf(x)\mathrm d\mu_n(x)=\int_Xf(x)\mathrm d\mu(x).$$

However, separability makes the things easier. For example, in a separable and complete metric space, each probability measure is tight (Ulam's theorem), and a weakly convergent sequence has to be uniformly tight if the space is complete (part of Prokhorov's theorem).