Exercise 2, § 3

Let X be a locally compact space in which a locally compact group G operates on the left continuously. Let E be a linear subspace of $\mathscr{M}(X)$, stable for the $\gamma(s)$ ($s \in G$), equipped with a quasi-complete locally convex topology finer than the topology of compact convergence in $\mathscr{K}(X)$. For every $s \in G$, let $\gamma_E(s)$ be the restriction of $\gamma(s)$ to E. Assume that $\mu \in E$ implies $|\mu| \in E$, and that the representation $\gamma_E$ of G in E is equicontinuous. Then, if $\mu \in E$ and $\nu \in \mathscr{M}^1(G)$, $\nu$ and $\mu$ are convolvable and $\nu * \mu = \gamma_E(\nu)\mu \in E$. (Make use notably of Prop. 17 of Ch. VI, §1, No. 7.)