Example

— Let X = \mathbf{R} and let μ be Lebesgue measure on \mathbf{R} (Ch. III, §1, No. 3); we are going to determine the outer measure of an open interval G = ]a, b[ (−∞ ≤ a < b ≤ +∞). Suppose first that a and b are finite. For every function f in $\mathscr{K}_+$ such that f ≤ φ_G, we have, by the theorem of the mean (FRV, II, §1, No. 5, Prop. 6),

whence $\mu^*(G) \leq b - a$. On the other hand, for every $\varepsilon > 0$ there exists a function $f \in \mathscr{K}_+$ such that $f \leq \varphi_G$ and $f(x) = 1$ for $a + \varepsilon \leq x \leq b - \varepsilon$, whence $\mu^*(G) \geq b - a - 2\varepsilon$; since $\varepsilon$ is arbitrary, $\mu^*(G) = b - a$; in other words, the outer measure of $G$ is equal to its length. This result extends at once to the case that $G$ is an unbounded open interval, since it then contains bounded open intervals of arbitrarily large length; thus $\mu^*(G) = +\infty$ in this case.

Now let $G$ be any open set in $\mathbf{R}$; $G$ is the union of a countable set (finite or infinite) of pairwise disjoint open intervals $]a_k, b_k[$ (GT, IV, §2, No. 5, Prop. 2), consequently

(Prop. 8); in other words: