Exercise 30, § 1

Let $G$ be a locally compact group, $\mu$ a left Haar measure on $G$, $\nu$ a bounded measure on $G$. Suppose that the mapping $s \to \gamma(s)\nu$ from $G$ into the Banach space $\mathscr{M}^1(G)$ is continuous. Now prove that the measure $\nu$ is of basis $\mu$. (Let $A$ be a compact $\mu$-negligible subset of $G$. By reasoning as for prop. 11, prove that $\nu(xA) = 0$ for $x$ running through an everywhere dense subset of $G$. Deduce that $v(A) = 0$.

Conversely, if $v$ is of basis $\mu$, the mapping $s \to \gamma(s)v$ from $G$ into $\mathscr{M}^1(G)$ is continuous: cf. Chapter VIII, § 2, No. 5, Prop. 8.