Proposition 6

Suppose that the locally compact group G is generated by a compact neighborhood of e. Then the set $\Gamma_c^0$ of measures $\alpha \in \Gamma^0$ such that $G/H_\alpha$ is compact is open in $\Gamma^0$, and the restriction to $\Gamma_c^0$ of the mapping $\alpha \mapsto \| \mu_\alpha \|$ is vaguely continuous.

With the notations of Prop. 5 of No. 2, we have, for $g \in \mathscr{K}_+(G)$,

For, if K is the support of $g$, the relation $\int g(xs)\, d\alpha(s) \geq 1$ for all $x \in G$ implies $KH_\alpha = G$, the integral obviously being zero on the complement of $KH_\alpha$, therefore $G/H_\alpha = \pi_\alpha(K)$ is compact. Given a measure $\alpha \in \Gamma_c^0$, it will therefore suffice to define a function $g \in \mathscr{K}_+(G)$ such that $\Gamma^0(g)$ is a neighborhood of $\alpha$ in $\Gamma^0$. Since $G/H_\alpha$ is compact and the canonical mapping $f \mapsto f_\alpha$ of $\mathscr{K}_+(G)$ into $\mathscr{K}_+(G/H_\alpha)$ is surjective (Ch. VII, §2, No. 2), there exists a function $g \in \mathscr{K}_+(G)$ such that $\int g(xs)\, d\alpha(s) = 2$ for every $x \in G$. Let K be the (compact) support of $g$, L a symmetric compact neighborhood of e in G that generates G; the mapping $\beta \mapsto g * \beta$ of $\mathscr{M}_+(G)$ into $\mathscr{C}(G)$ being vaguely continuous (§ 4, No. 2, Remark 1), there exists a neighborhood W of $\alpha$ in $\Gamma^0$ such that

for all $\beta \in W$ and $x \in LK$. The first member of (7) being equal to zero outside $KH_\beta$, the relation $\beta \in W$ implies

from which one deduces, by induction on $n$, that $L^n K \subset KH_\beta$ for every integer $n > 0$; since L generates G, we therefore have $G = KH_\beta$ for every measure $\beta \in W$, which proves that $W \subset \Gamma_c^0$. On the other hand, the first member of (7) being invariant on the right under $H_\beta$, the inequality (7) is also valid for $x \in LKH_\beta = G$; therefore $W \subset \Gamma^0(g)$, which proves the proposition.

Finally, (iii) will be a consequence of the following proposition: