Corollary 1

— For every compact subset K of X and every number $a > 0$, the set B of measures $\mu$ on X such that $\operatorname{Supp}(\mu) \subset K$ and $\| \mu \| \leq a$ is an equicontinuous subset of $\mathscr{C}'(X; \mathbf{C})$ that is compact for the topology $\mathscr{T}$ of compact convergence.

For, let H be a subset of $\mathscr{C}(X; \mathbf{C})$ consisting of functions that are uniformly bounded on K; there exists a number $c > 0$ such that $|\mu(f)| \leq c \cdot \| \mu \| \leq ac$ for every function $f \in H$ and every measure $\mu \in B$, by virtue of (22); therefore $B \subset acH^\circ$ in the dual $\mathscr{C}'(X; \mathbf{C})$ of $\mathscr{C}(X; \mathbf{C})$, which proves the equicontinuity of B; the fact that B is compact for $\mathscr{T}$ follows from the fact that, on B, $\mathscr{T}$ and the vague topology induce the same topology (Prop. 16 and Ch. III, §1, No. 10, Prop. 17) and the fact that B is vaguely compact (Ch. III, §1, No. 9, Cor. 2 of Prop. 15 and §2, No. 2, Prop. 6).