Proposition 11

— Every measure with compact support is bounded.

For, $|\mu|$ is also a measure with compact support, thus we can restrict attention to the case that $\mu \geq 0$; if $h$ is a continuous mapping of $X$ into $[0, 1]$, with compact support and equal to 1 on $\mathrm{Supp}(\mu)$, then for every function $f \in \mathscr{K}(X; \mathbf{C})$ one has $|f(x)| \leq \|f\| h(x)$ on $\mathrm{Supp}(\mu)$, therefore (Cor. 2 of Prop. 8) $\mu(|f|) \leq \mu(h)\|f\|$, which proves the proposition (§ 1, No. 8).