Proposition 7

— If $\Phi$ is a filter on $\mathscr{M}(X; \mathbf{C})$ such that the support of $\mu$ recedes indefinitely along $\Phi$, then $\mu$ converges vaguely to 0 with respect to $\Phi$.

Let $f$ be any function in $\mathscr{K}(X; \mathbf{C})$ and let $K$ be its support. By hypothesis, there exists a set $M \in \Phi$ such that for every measure $\mu \in M$, $\operatorname{Supp}(\mu) \cap K = \varnothing$; it follows that $\mu(f) = 0$ for all $\mu \in M$, which proves the proposition.