Proposition 5

Tag 08PC, permanent. Cite it as [Bourbaki, Tag 08PC].

— The outer measure of a relatively compact open set $G$ is finite.

For, there exists in this case a function $f \in \mathscr{K}_+$ such that $\varphi_G \leq f$ (Ch. III, §1, No. 2, Lemma 1), whence

\mu^*(G) = \mu^*(\varphi_G) \leq \mu^*(f) = \mu(f) < +\infty.

An open set of finite outer measure is not always relatively compact (Exer. 3).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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