Corollary 1

Let $G$ be a locally compact group, $\beta$ a Haar measure on $G$, $E$ a Hausdorff barreled locally convex space, $U$ a linear representation of $G$ in $E$, such that $U(s) \in \mathscr{L}(E;E)$ for all $s \in G$, $\beta$-measurable when $\mathscr{L}(E;E)$ is equipped with the topology of pointwise convergence. Then $U$ is a continuous linear representation.

Let $\Gamma$ be the group of automorphisms of $E$, equipped with the topology of pointwise convergence. This topology is Hausdorff and is invariant under translations. Let $K$ be a compact subset of $\Gamma$. Then $K$ is bounded in $\mathscr{L}(E;E)$ equipped with the topology of pointwise convergence, hence is equicontinuous (TVS, III, §4, No. 2, Th. 1); therefore the mapping $(u,v) \mapsto v \circ u$ of $K \times K$ into $\mathscr{L}(E;E)$ is continuous (loc. cit., §5, No. 5, Cor. 1 of Prop. 9). Therefore, for every $x \in E$, the mapping $s \mapsto U(s)x$ of $G$ into $E$ is continuous (Prop. 18). Since $E$ is barreled, $U$ is continuous (§2, No. 1, Prop. 1).