Exercise 30, § 1

Let $G$ be a locally compact group, $\mu$ a left Haar measure on $G$, and $\nu$ a bounded measure on $G$. Assume that the mapping $s \mapsto \gamma(s)\nu$ of $G$ into the Banach space $\mathscr{M}^1(G)$ is continuous. Show that the measure $\nu$ has base $\mu$. (Let $A$ be a $\mu$-negligible compact subset of $G$. Arguing as for Prop. 11, show that $\nu(xA) = 0$ for $x$ running over a dense subset of $G$. From this, deduce that $\nu(A) = 0$.)

Conversely, if $\nu$ has base $\mu$, then the mapping $s \mapsto \gamma(s)\nu$ of $G$ into $\mathscr{M}^1(G)$ is continuous: cf. Ch. VIII, §2, No. 5, Prop. 8.