Corollary

— Let T be a subspace of a completely regular space X, and let H be a subset of $\mathscr{M}^b(T)$ that satisfies Prokhorov’s condition. If i denotes the canonical injection of T into X, then the restriction to H of the mapping $\mu \mapsto i(\mu)$ of $\mathscr{M}^b(T)$ into $\mathscr{M}^b(X)$ is a homeomorphism of H onto its image.

It suffices to treat the case that H is closed (Prop. 11), hence compact; the conclusion then follows from the fact that $\mu \mapsto i(\mu)$ is continuous and injective.

Recall that this result is also valid for an arbitrary subset of $\mathscr{M}_+^b(T)$ (No. 3, Prop. 8).