Proposition 4

— Let $\Gamma^0$ be the set of measures $\alpha \in \Gamma$ such that $H_\alpha$ is unimodular, and for every $\alpha \in \Gamma^0$ set $\mu_\alpha = \mu / \alpha$; then the mapping $\alpha \mapsto \| \mu_\alpha \|$ of $\Gamma^0$ into $\overline{\mathbf{R}}$ is lower semi-continuous for the vague topology.

For every $\alpha \in \Gamma^0$ and every function $f \in \mathscr{K}(G)$, set

where the convolution product is taken relative to the right Haar measure $\mu$ and where one makes use of the fact that $\dot{\alpha} = \alpha$ (§ 4, No. 4, formula (11)). We know (Ch. VII, §2, No. 1, Prop. 2) that the mapping $f \mapsto f_\alpha$ of $\mathscr{K}_+(G)$ into $\mathscr{K}_+(Q_\alpha)$ is surjective; therefore, by (2),

where one has set

To establish the proposition, it will suffice to show that, given $f \in \mathscr{H}_+(G)$, the mapping $\alpha \mapsto \|f_\alpha\|$ is vaguely continuous. Now, let K be the support of $f$; the function $f * \alpha$ has its support contained in $KH_\alpha$ and is invariant on the right under $H_\alpha$; consequently

The conclusion therefore follows from the fact that the mapping $\alpha \mapsto f * \alpha$ of $\mathscr{M}_+(G)$ equipped with the vague topology, into $\mathscr{C}(G)$ equipped with the topology of compact convergence, is continuous (§ 4, No. 2, Remark 1).

Recall that if, for a measure $\alpha \in \Gamma^0$, $\|\mu_\alpha\|$ is finite, then G is necessarily unimodular (Ch. VII, §2, No. 6, Cor. 3 of Th. 3).