Proposition 7

We retain the notations of Prop. 6. If each $\mu_\alpha$ is a left (resp. right) Haar measure on $G_\alpha$, then $\mu$ is a left (resp. right) Haar measure on $G$.

Suppose, for example, that the $\mu_\alpha$ are left Haar measures. Let $s \in G$. For every $x \in G$,

therefore $\varphi_\alpha(\gamma(s)\mu) = \gamma(\varphi_\alpha(s))\mu_\alpha = \mu_\alpha$. Therefore $\gamma(s)\mu = \mu$ by Lemma 2 b), thus $\mu$ is a left Haar measure.

We suppose henceforth the $K_\alpha$ to be not only compact, but open in $G$. The $G_\alpha$ are then discrete and, for $\beta \geq \alpha$, $K_\alpha/K_\beta$ is a compact and discrete group, hence is finite. The group $G$ is unimodular (Prop. 3).