$(\mu_n)$ possesses a weakly convergent subsequence.

Question

Let $X$ be a compact metric space and let $(\mu_n)$ be a sequence of probability measures on $X$. Prove that $(\mu_n)$ possesses a weakly convergent subsequence.

Require Hints to proceed with the problem.

Answer 1

Hint: you need two theorems to prove this.

1. Riesz Representation Theorem: Suppose that $X$ is a compact metric space and suppose that $T:C(X,\mathbb R) \to \mathbb R$ is a positive linear functional on $C(X,\mathbb R) = \{f: X \to \mathbb R \,: \, f \text{ is continuous}\}$. Then there is a unique (positive) Radon measure $\mu$ on $X$ such that $$T(f) = \int_X f \, d\mu, \,\,\,\,\, \forall f \in C(X,\mathbb R).$$

2. Banach-Alaoglu Theorem. If $V$ is a separable normed vector space with continuous dual $V^*$, then the unit ball in $V^*$ is sequentially compact in the weak$^*$ topology.