Consider the set $\mathcal M (\mathbb R)$ of all (Borel) probability measures on $\mathbb R$ corresponding to distributions of real-valued random variables. Prokhorov's theorem says that a subset of $\mathcal M (\mathbb R)$ is compact with respect to the topology of weak convergence if and only if it is tight. Since $\mathcal M (\mathbb R)$ itself isn't tight, it is not compact.
However, it seems that the set of subprobabilities $\bar{\mathcal{M}}(\mathbb{R})$, defined to be all measures integrating to at most 1, is a compactification of $\mathcal M (\mathbb R)$. That is, it is compact with respect to a topology whose subset topology on $\mathcal M (\mathbb R)$ is the topology of weak convergence. It is suggested in Chen (2017), Kiefer-Wolfowitz 1956, and here, though without further proof.
Is there a relatively elementary, intuitive exposition of this result?
Best Answer
Yes, consider $K$, the one point compactification of the line. The space of probability measures on $K$ is compact, and is homeomorphic to the space of subprobability measures on $\mathbb R$.