Proposition 16

— Let $K$ be a compact subset of $X$, $H$ a vaguely bounded subset of $\mathscr{M}(X; \mathbf{C})$; then, the bilinear form $(f, \mu) \mapsto \langle f, \mu \rangle$ is continuous on $\mathscr{K}(X, K; \mathbf{C}) \times H$ when $\mathscr{K}(X, K; \mathbf{C})$ is equipped with the topology of uniform convergence and $H$ with the vague topology.

For, there exists a number $M \geqslant 0$ such that

for every function $f \in \mathscr{K}(X, K; \mathbf{C})$ and every measure $\mu \in H$ (Prop. 15). If $\mu_0$ and $\mu$ are two measures belonging to $H$, $f_0$ and $f$ two functions in $\mathscr{K}(X, K; \mathbf{C})$, then and the last quantity is arbitrarily small when $\|f - f_0\|$ and $|\mu(f_0) - \mu_0(f_0)|$ are, which proves the proposition.