Proposition 2

Tag 08S3, permanent. Cite it as [Bourbaki, Tag 08S3].

Integration, chapter IV, § 4. Integrable functions and sets

— For every integrable function $f$ ,

\left| \int f \, d\mu \right| \leq \int |f| \, d|\mu|.

This follows at once from the inequality (1) by passage to the limit, on taking into account (3) and the continuity of $N_1(f)$ on $\mathscr{L}_F^1$ .

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

Cited by

Lemma 1, § 5

Cites

Nothing.