Proposition 6

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

Integration, chapter IV, § 2. Negligible functions and sets

— If $f$ and $g$ are two functions $\geqslant 0$ (finite or not) defined on $X$ such that $f(x) = g(x)$ almost everywhere, then $|\mu|^*(f) = |\mu|^*(g)$ .

Let $N$ be the negligible set of points $x \in X$ such that $f(x) \neq g(x)$ . The functions $\inf(f, g)$ and $\sup(f, g)$ being equal except at the points of $N$ , it suffices to prove the proposition assuming $f \leqslant g$ . Let $h$ be the function equal to $+\infty$ at the points of $N$ , and to 0 on $CN$ ; then $f \leqslant g \leqslant f + h$ , thus

|\mu|^*(f) \leqslant |\mu|^*(g) \leqslant |\mu|^*(f + h) \leqslant |\mu|^*(f) + |\mu|^*(h) = |\mu|^*(f)

(since $h$ is negligible), whence the proposition.

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

Cited by

Proposition 10, § 2
Proposition 3, § 3
Corollary 1, § 4
Proposition 7, § 1
Proposition 2, § 3

Cites

Nothing.