Exercise 8, § 3

Let G be a locally compact group; denote by $\mathscr{C}'_+(G)$ the set of positive measures on G with compact support. Show that the mapping $(\mu, \nu) \mapsto \mu * \nu$ of $\mathscr{M}_+(G) \times \mathscr{C}'_+(G)$ into $\mathscr{M}_+(G)$ is continuous when $\mathscr{C}'(G)$ is equipped with the weak topology $\sigma(\mathscr{C}'(G), \mathscr{C}(G))$, and $\mathscr{M}(G)$ with the vague topology $\sigma(\mathscr{M}(G), \mathscr{K}(G))$. (Make use of Exer. 10 of Ch. III, §1 and the fact that G is paracompact.)