Definition 2

Let T be a topological space, and let H be a subset of $\mathscr{M}^b(T)$; one says that H satisfies Prokhorov’s condition if a) $\sup_{\mu \in H} |\mu|(1) < +\infty$; b) for every number $\varepsilon > 0$, there exists a compact subset $K_\varepsilon$ of T such that

It can be shown that if T is completely regular, the set of conditions a) and b) is equivalent to the following condition: there exists a real function $f \geq 1$ on T, such that the set of points t of T satisfying $f(t) \leq c$ is compact for every $c \in \mathbf{R}_+$ (which in particular implies that $f$ is lower semi-continuous), and such that $\sup_{\mu \in H} |\mu|(f) < +\infty$. Moreover, when T is locally compact, one obtains an equivalent statement by requiring f to be continuous (cf. Exer. 10).