Corollary 3

*Let $G$ be a locally compact group, $E$ a quasi-complete locally convex space, and $U$ a continuous linear representation of $G$ in $E$. Let $\beta$ be a positive measure on $G$, $a$ an element of $G$, and $\mathscr{B}$ a base for the filter of neighborhoods of $a$, formed of compact neighborhoods. For every $V \in \mathscr{B}$, let $f_V$ be a continuous function $\geqslant 0$ on $G$, with support contained in $V$, and such that $\int f_V\, d\beta = 1$. Then, for every $x \in E$,

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

The mapping $s \mapsto U(s)x$ of $G$ into $E$ is continuous. By Cor. 2, $U(a)x = \lim_V \int (U(s)x) \cdot f_V(s)\, d\beta(s)$ with respect to the section filter of $\mathscr{B}$, that is, $U(a)x = \lim_V U(f_V \cdot \beta)x$.