Proposition 19

— For every subset A of X, $\mu^*(A)$ is the infimum of the outer measures of the open sets containing A.

The proposition is obvious if $\mu^*(A) = +\infty$. In the contrary case, for every $\varepsilon$ such that $0 < \varepsilon < 1$, there exists a function $f \in \mathscr{J}_+$ such that $\varphi_A \leq f$ and $\mu^*(A) \leq \mu^*(f) \leq \mu^*(A) + \varepsilon$. Let G be the set of $x \in X$ such that $f(x) > 1 - \varepsilon$. Since $f$ is lower semi-continuous, G is open (GT, IV, §6, No. 2, Prop. 1) and contains A; on the other hand $f \geq (1 - \varepsilon)\varphi_G$, whence

since $\varepsilon$ is arbitrary, we see that $\mu^*(G)$ differs as little as we please from $\mu^*(A)$, whence the proposition.