Proposition 7

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

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

— Let $\mathfrak{B}$ be a set of open subsets of X that is directed for the relation ⊂; then

\mu^*\left(\bigcup_{G \in \mathfrak{B}} G\right) = \sup_{G \in \mathfrak{B}} \mu^*(G).

The functions φ_G form a directed set (for ≤) in $\mathscr{I}_+$ and their upper envelope is the characteristic function of the union of the sets G ∈ $\mathfrak{B}$ ; the proposition is thus a consequence of Th. 1.

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

Cited by

Proposition 11, § 4
Corollary 2, § 1
Corollary, § 1
Remark, § 2

Cites

Th. 1