Corollary 3

Tag 090L, permanent. Cite it as [Bourbaki, Tag 090L].

Integration, chapter V, § 2. Summable families of positive measures

— For every function $f \in \mathscr{F}_+(X)$ ,

$\nu^*(f) \geq \sum_{\alpha \in A} \lambda_\alpha^*(f).$

The inequality is obvious if $f$ is not $\nu$ -moderated, because then $\nu^*(f) = +\infty$ ( $§ 1$ , No. 2, Prop. 7). If $f$ is $\nu$ -moderated then $f$ is $\lambda_\alpha$ -moderated for every $\alpha \in A$ , because every $\nu$ -integrable open set is $\lambda_\alpha$ -integrable; the relation (4) then follows at once from (3) and from Prop. 7 of $§ 1$ , No. 2.

It can happen that the two members of (4) are not equal, even when $A$ is countable and each of the $\lambda_\alpha$ is a point measure ( $§ 1$ , Exer. 4 a)).

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

Cited by

Nothing.

Cites

Nothing.