Proposition 11

Let T be a completely regular space, and let H be a subset of $\mathscr{M}^b(T)$ that satisfies Prokhorov’s condition; then its closure $\overline{H}$ in $\mathscr{M}^b(T)$ satisfies Prokhorov’s condition.

For, the functions $\mu \mapsto |\mu|^*(1)$, $\mu \mapsto |\mu|^*(T - K_\varepsilon)$ are lower semi-continuous on $\mathscr{M}^b(T)$ by Prop. 6 of No. 3.

The interest of Prokhorov’s condition comes from the following theorem, whose converse will be studied later on (Th. 2).