Remark

The hypothesis (iii) of Prop. 2 is satisfied notably when $\mu$ is bounded. For, for every $n > 0$, there then exists a compact subset $K_n$ of $G$ such that

(Ch. IV, §4, No. 7), and $\mu$ is carried by the union of the $K_n$. More generally, let $\rho$ be a lower semi-continuous finite function $> 0$ on $G$ such that $\rho(st) \leq \rho(s)\rho(t)$; if $\mu \in \mathscr{M}^\rho$, the hypothesis (iii) is satisfied; for, $\rho \cdot \mu$ is bounded, and $\mu$ is carried by the same subsets as $\rho \cdot \mu$ since, on every compact subset of $G$, $\rho$ is bounded below by a constant $> 0$.