Proposition 15

— Let $H$ be a subset of $\mathscr{M}(X; \mathbf{C})$. The following properties are equivalent:

a) $H$ is vaguely bounded. b) $H$ is vaguely relatively compact. c) $H$ is equicontinuous. d) For every compact subset $K$ of $X$, there exists a number $M_K \geq 0$ such that $|\mu(f)| \leq M_K \|f\|$ for every measure $\mu \in H$ and every function $f \in \mathscr{K}(X, K; \mathbf{C})$.

Since $\mathscr{K}(X; \mathbf{C})$ is a barreled space (No. 1, Prop. 2), the equivalence of properties a), b) and c) follows from TVS, III, §4, No. 1, Scholium.

It is clear that d) implies a). Finally, if $H$ is equicontinuous then the set of restrictions of the measures $\mu \in H$ to $\mathscr{K}(X, K; \mathbf{C})$ is also equicontinuous, whence the condition d), since $\mathscr{K}(X, K; \mathbf{C})$ is a normed space.

\* We shall see in Ch. IV, §4, No. 6 that the conditions of Proposition 15 are also equivalent to the condition that, for every compact subset $K$ of $X$, there exists a constant $M_K$ such that $|\mu|(K) \leq M_K$ for every measure $\mu \in H$.*