Proposition 19

— *Let $G$ be a locally compact group, $\beta$ a relatively invariant positive measure $\neq 0$ on $G$, $\mathscr{B}$ a base for the filter of neighborhoods of $e$ in $G$, consisting of compact neighborhoods. For every $V \in \mathscr{B}$, let $f_V$ be a continuous function $\geq 0$ on $G$, with support contained in $V$, such that $\int f_V d\beta = 1$. If $\mu$ is a measure on $G$ then, in $\mathscr{M}(G)$ equipped with the topology of compact convergence in $\mathscr{H}(G)$,

the limit being taken with respect to the section filter of $\mathscr{B}$.

For the topology of compact convergence in $\mathscr{C}(G)$, $f_V \cdot \beta$ tends to $\varepsilon_e$ with respect to the section filter of $\mathscr{B}$ (§ 2, No. 7, Cor. 1 of Lemma 4). Therefore $\mu = \lim_V \mu * (f_V \cdot \beta) = \lim_V (f_V \cdot \beta) * \mu$ in $\mathscr{M}(G)$ equipped with the topology of compact convergence in $\mathscr{H}(G)$ (§ 3, No. 3, Cor. of Prop. 12).