Proposition 10

— Let $\mu$ be a measure on $G$, $\nu$ a measure on $X$, $\mu$ and $\nu$ being convolvable. Suppose in addition that there exists a positive measure $\beta$ on $X$ such that $\gamma(s)\nu$ has base $\beta$ for every $s \in G$. Then $\mu * \nu$ has base $\beta$. Let $K$ be a $\beta$-negligible compact subset of $X$. Then $K$ is $\gamma(s)|\nu|$-negligible for every $s \in G$. Now,

(§1, No. 5, Prop. 7), and the mapping $s \mapsto \varepsilon_s * |\nu|$ is vaguely continuous (§2, Prop. 6). Therefore $K$ is $|\mu| * |\nu|$-negligible by Ch. V, §3, No. 3, Th. 1. Therefore $|\mu| * |\nu|$ has base $\beta$ (Ch. V, §5, No. 5, Th. 2).