Corollary 1

Tag 0931, permanent. Cite it as [Bourbaki, Tag 0931].

Integration, chapter V, § 5. Measures defined by numerical densities

— Let $\mathscr{N}$ be a set of positive measures with base $\mu$ , admitting a supremum $\nu$ in $\mathscr{M}(T)$ ; then $\nu$ is a measure with base $\mu$ .

The Cor. of Prop. 2 permits reduction to the case that $\mathscr{N}$ is an increasing directed set. For every locally $\mu$ -negligible set $A$ one then has, by Prop. 11 of §1, No. 4,

\nu^\bullet(A) = \sup_{\lambda \in \mathscr{N}} \lambda^\bullet(A) = 0.

Theorem 2 therefore implies that $\nu$ is a measure with base $\mu$ .

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

Cited by

Corollary 2, § 5

Cites

Prop. 11 of §1, No. 4
Cor. of Prop. 2
Theorem 2