Exercise 21, § 4

Let X be a locally compact space, $f$ a numerical function $\geqslant 0$ defined on X. In order that the mapping $\mu \mapsto \mu^*(f)$ of $\mathscr{M}_+(X) \cap \mathscr{C}'(X; \mathbf{C})$ into $\overline{\mathbf{R}}$ be continuous for the topology of compact convergence, it is necessary and sufficient that $f$ be continuous on X.