Proposition 13

— Let T be a locally compact space, and H a bounded subset of the normed space $\mathscr{C}^b(T; \mathbf{C})$. The mapping $(\mu, f) \mapsto \mu(f)$ of $\mathscr{M}_+^b(T) \times H$ into $\mathbf{C}$ is then continuous, when $\mathscr{M}_+^b(T)$ is equipped with the tight topology, and H with the topology of compact convergence.

Let $\mu \in \mathscr{M}_+^b(T), f \in H$, and let M be a real number such that $||\mu|| < M$, and $|g| \leq M$ for all $g \in H$. Let $\varepsilon$ be a number $> 0$ and choose a compact subset K of T such that $\mu^*(T - K) < \varepsilon$, then a relatively compact open neighborhood S of K. The set U of measures $\lambda \in \mathscr{M}_+^b(T)$ satisfying the inequalities

is then a neighborhood of $\mu$ in $\mathscr{M}_+^b(T)$ (No. 3, Prop. 6). In addition, let V be the neighborhood of $f$ in H consisting of the functions $g \in H$ such that

Let $\lambda \in U$ and $g \in V$; since the function $|g - f|$ is bounded above by $\varepsilon$ in $S$, and by $2M$ in $T - S$, we have

from which one deduces This proves the continuity of the mapping $(\lambda, g) \mapsto \lambda(g)$ at the point $(\mu, f)$ of $\mathscr{M}_+^b(T) \times H$.