Corollary

— With hypotheses and notations as in Prop. 9, the restriction of the mapping $\mu \mapsto \int f\, d\mu$ to the set $\mathscr{M}_+(X)$ of positive measures, or to a vaguely bounded subset B of $\mathscr{M}(X; \mathbf{C})$, is vaguely continuous.

For, it follows from § 1, No. 10, Props. 17 and 18 that, on $\mathscr{M}_+(X)$ or on $B$, the topology induced by the topology of strictly compact convergence is the same as the topology induced by the vague topology.

However, the mapping $\mu \mapsto \int f\, d\mu$ is not necessarily continuous on all of $\mathscr{M}(X; \mathbf{C})$ for the vague topology (Exer. 2).