Remark

Let $T$ be a completely regular space, $M$ a subset of $\mathscr{M}^b(T)$ that satisfies Prokhorov’s condition, $H$ a bounded subset of $C^b(T)$. An argument very close to the one just made may be used to prove that the mapping $(\lambda, g) \mapsto \lambda(g)$ of $M \times H$ into $\mathbf{C}$ is continuous when $M$ is equipped with the tight topology and $H$ with the topology of compact convergence.