Prokhorov's theorem

Remarks

Tight means that for every $\varepsilon > 0$ there is a compact set $K$ with $\mu(K) > 1 - \varepsilon$ for all $\mu \in \mathcal{M}$. For a compact space $X$, part 1 follows from the Riesz representation theorem and the weak-* compactness of the unit ball of the dual of $C(X)$. The general case reduces to compact sets by tightness. For $X = \R$, part 1 is close to Helly's selection theorem for distribution functions.

Example: the normal distributions $N(m, \sigma^2)$ with $\abs{m} \le 1$ and $0 < \sigma \le 1$ form a tight family, using $K = [-M, M]$ with $M$ large. So every sequence of them has a weakly convergent subsequence. The family of Dirac measures $\delta_n$, $n \in \N$, on $\R$ is not tight, and no subsequence converges weakly. The mass escapes to infinity.

Going deeper

For a student

In probability, one proves convergence in distribution in two steps. First one shows tightness, which gives limit points by part 1. Then one shows that all limit points are the same measure, for example through characteristic functions or finite-dimensional distributions. This scheme proves Donsker's invariance principle for random walks. On a Polish space, the set of probability measures with the weak topology is itself a Polish space, metrized by the Lévy–Prokhorov metric.