Definition 1

Tag 08P3, permanent. Cite it as [Bourbaki, Tag 08P3].

Integration, chapter IV, § 1. Upper integral of a positive function

Given a positive measure $\mu$ on $X$ , one calls upper integral of a function $f \in \mathscr{J}_+$ (with respect to $\mu$ ) the positive number (finite or equal to $+\infty$ )

\mu^*(f) = \sup_{g \in \mathscr{H}_+, g \leq f} \mu(g).

For every function $f \in \mathscr{H}_+$ , it is clear that $\mu^*(f) = \mu(f)$ , in other words $\mu^*$ is an extension of $\mu$ to $\mathscr{J}_+$ .

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

Cited by

Definition 3, § 1
Proposition 2, § 1
Theorem 1, § 1
Proposition 5, § 4
Lemma 4, § 5

Cites

Nothing.