Exercise 10, § 1

Let $\Phi$ be a filter on the set $\mathscr{M}_+(X)$ of positive measures on $X$. Show that if $\Phi$ is vaguely convergent then, for every compact subset $K$ of $X$, there exist a set $M \in \Phi$ and a number $a_K > 0$ such that $|\mu(f)| \leq a_K \cdot \|f\|$ for every function $f \in \mathscr{H}(X, K; \mathbf{C})$ and every measure $\mu \in M$.