Proposition 4

*Let G be a locally compact group, $\Gamma$ a topological group, and $\gamma \mapsto u_\gamma$ a homomorphism of $\Gamma$ into the group $\mathscr{G}$ of automorphism of G, such that $(\gamma, x) \mapsto u_\gamma(x)$ is a continuous mapping of $\Gamma \times G$ into G. Then, the mapping $\gamma \mapsto \operatorname{mod}(u_\gamma)$ is a continuous representation of $\Gamma$ in $\mathbf{R}_+^*$.

This mapping is obviously a representation (algebraic) of $\Gamma$ in $\mathbf{R}_+^*$; it will suffice to prove its continuity. Let $f \in \mathscr{K}(G)$ and let S be its support.*

Let $\gamma_0 \in \Gamma$ and let $U$ be a relatively compact neighborhood of $u_{\gamma_0}^{-1}(S)$. The mapping $\gamma \mapsto u_\gamma$ is a continuous mapping of $\Gamma$ into $\mathscr{G}$ equipped with the topology of compact convergence (GT, X, §3, No. 4, Th. 3); therefore $u_\gamma^{-1}(S) \subset U$ for $\gamma$ sufficiently near $\gamma_0$. Lemma 1 of No. 1 then proves that $\int f(u_\gamma(x)) d\mu(x)$ (where $\mu$ denotes a left Haar measure of $G$) depends continuously on $\gamma$; whence the proposition.