Theorem 1.6.2 (Prokhorov).label Let $X$ be a Polish space and $\mathcal{P}\subset \mathbf{M}_{1}(X, \cb(X))$, then the following are equivalent:

1. (1)

   For each $\alpha \in (0, 1)$, there exists $K \subset X$ compact such that

   \[\inf_{\mu \in \mathcal{P}}\mu(K) \ge \alpha\]

   
   
2. (2)

   $\mathcal{P}$ is weakly precompact.