Corollary

The mapping $(\lambda, \mu) \mapsto \lambda * \mu$ of $\mathscr{C}'(G) \times \mathscr{M}(X)$ into $\mathscr{M}(X)$ is hypocontinuous relative to the equicontinuous subsets of $\mathscr{C}'(G)$ and the compact subsets of $\mathscr{M}(X)$ ($\mathscr{C}'(G)$ and $\mathscr{M}(X)$ being equipped with the topology of compact convergence in $\mathscr{C}(G)$ and $\mathscr{H}(X)$, respectively).

For, $\mathscr{M}(X)$, equipped with the topology of compact convergence in $\mathscr{H}(X)$, is quasi-complete. Therefore the mapping $(\lambda, \mu) \mapsto \gamma(\lambda)\mu$ of $\mathscr{C}'(G) \times \mathscr{M}(X)$ into $\mathscr{M}(X)$ is hypocontinuous relative to the equicontinuous subsets of $\mathscr{C}'(G)$ and the compact subsets of $\mathscr{M}(X)$ (§2, No. 6). It then suffices to apply Prop. 12.