Corollary 1

— Let $K$ be a compact subset of $G$, $V$ a symmetric compact neighborhood of $e$ in $G$, $c$ a real number $> 0$. The restriction of the mapping $\alpha \mapsto \| \mu_\alpha \|$ to the set of $\alpha \in \Gamma^0$ such that $G = KH_\alpha$ and $\alpha(V) \geq c$ is vaguely continuous.

For, let $g \in \mathscr{K}_+(G)$ be a function such that $g(x) \geq 1/c$ for $x \in KV$. For every $x \in K$,

for $\alpha$ satisfying the conditions of the statement; since, moreover, $\pi_\alpha(K) = Q_\alpha$, one has $\alpha \in \Gamma^0(g)$, whence the corollary.