Lemma 3

Let E be a set filtered by a filter $\Phi$, and let $\xi \mapsto \alpha(\xi)$ be a mapping of E into $\Gamma$ that converges vaguely with respect to $\Phi$ to a measure $\alpha_0 \neq 0$. On the other hand, let $\xi \mapsto t_\xi$ be a mapping of $E$ into $G$ such that $t_\xi \in H_{\alpha(\xi)}$ for every $\xi \in E$. If $s$ is a cluster point of the mapping $\xi \mapsto t_\xi$ with respect to $\Phi$, then $\delta(s)\alpha_0 = \alpha_0$.

Replacing if necessary $\Phi$ by a finer filter, we can suppose that $s$ is a limit of $\xi \mapsto t_\xi$ with respect to $\Phi$; by Lemma 1, $\delta(t_\xi)\alpha(\xi) = \alpha(\xi)$ for every $\xi \in E$, and the conclusion follows from the continuity of the mapping $(u, \lambda) \mapsto \delta(u)\lambda$ on $G \times \mathscr{M}_+(G)$ (§ 3, No. 3, Prop. 13).

To prove Prop. 1 it suffices, by Lemma 1, to show that if a filter $\Psi$ on $\Gamma$ converges vaguely to a measure $\alpha_0 \neq 0$ and if $s$ belongs to the support of $\alpha_0$, then $\delta(s)\alpha_0 = \alpha_0$. Now, for every neighborhood $V$ of $s$ in $G$, there exists an $M \in \Psi$ such that, for every $\alpha \in M$, one has $V \cap H_\alpha \neq \varnothing$, by Lemma 2. For every neighborhood $V$ of $s$ and every $\alpha \in \Gamma$, let $t_{V,\alpha}$ be a point of $V \cap H_\alpha$ if $V \cap H_\alpha \neq \varnothing$, and any point of $H_\alpha$ in the contrary case; if $\Theta$ is the section filter of the filter of neighborhoods of $s$, and $\Phi$ is the product filter $\Theta \times \Psi$, then $s$ is, by the foregoing, a cluster point of $(V, \alpha) \mapsto t_{V,\alpha}$ with respect to $\Phi$. Since, on the other hand, the mapping $(V, \alpha) \mapsto \alpha$ has $\alpha_0$ as limit with respect to $\Phi$, the proposition follows from Lemma 3.