Corollary 4

— For every open set $U$ in $X$, $|\mu|^*(U)$ is the supremum of the measures $|\mu|(K)$ of the compact sets $K \subset U$.

If $|\mu|^*(U) < +\infty$, this is immediate from Th. 4. The following argument covers also the case that $|\mu|^*(U) = +\infty$. Since $X$ is locally compact and $U$ is open, $\varphi_U$ is the upper envelope of the set $H$ of functions $f \in \mathscr{K}_+$ such that $f \leq \varphi_U$ and $\operatorname{Supp}(f) \subset U$ (cf. the proof of §1, No. 1, Lemma), and, since $H$ is directed for $\leq$, we have $|\mu|^*(U) = \sup_{f \in H} |\mu|(f)$ by §1, No. 1, Th. 1; the corollary is then immediate from the fact that if $f \in H$ and $K = \operatorname{Supp}(f)$, then $f \leq \varphi_K \leq \varphi_U$.

Note that $|\mu|^*(U)$ is also the supremum of the measures $|\mu|(G)$ of the relatively compact open sets such that $\overline{G} \subset U$. For, if $K$ is a compact set contained in $U$ then, for every $x \in K$, there exists a relatively compact open neighborhood $V$ of $x$ such that $\overline{V} \subset U$. On covering $K$ by a finite number of these neighborhoods, their union $G$ is a relatively compact open set such that $\overline{G} \subset U$ and $K \subset G$, whence $|\mu|(K) \leq |\mu|(G) \leq |\mu|^*(U)$.