Exercise 5, § 2

Let $G$ be a locally compact group. For every $s \in G$, let $\psi_s$ be the automorphism of the additive group $\mathbf{R}$ defined by $\psi_s(x) = \Delta_G(s)x$. Let $\Gamma$ be the topological semi-direct product of $G$ and $\mathbf{R}$ defined by $s \mapsto \psi_s$ (GT, III, §2, No. 10). Show that $\Gamma$ is unimodular.