Lemma 2

Let X be a locally compact space and for every measure $\alpha \in \mathscr{M}_+(X) - \{0\}$, let $S_\alpha$ be the support of $\alpha$. Let $\Phi$ be a filter on $\mathscr{M}_+(X) - \{0\}$ that converges vaguely to a measure $\alpha_0 \neq 0$. Then, for every neighborhood V of a point s of the support of $\alpha_0$, there exists a set $M \in \Phi$ such that, for every $\alpha \in M$, one has $V \cap S_\alpha \neq \varnothing$. For, if $\varphi \in \mathscr{K}_+(X)$ is a function with support contained in V and such that $\int \varphi(x) d\alpha_0(x) > 0$, by definition there exists a set $M \in \Phi$ such that $\int \varphi(x) d\alpha(x) > 0$ for all $\alpha \in M$, which implies $V \cap S_\alpha \neq \varnothing$.