Proposition 4

Tag 08S8, permanent. Cite it as [Bourbaki, Tag 08S8].

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

— Let $(f_n)$ be an increasing (resp. decreasing) sequence of integrable numerical functions. For the upper (resp. lower) envelope $f$ of the sequence to be integrable, it is necessary and sufficient that $\sup_n \int f_n \, d|\mu| < +\infty$ (resp. $\inf_n \int f_n \, d|\mu| > -\infty$ ), in which case

\int f \, d\mu = \lim_{n \to \infty} \int f_n \, d\mu .

We limit ourselves to considering an increasing sequence. The sequence $g_n = f_n + f_1^-$ is increasing and consists of integrable functions $\geqslant 0$ ; since its upper envelope is $g = f + f_1^-$ , the proposition follows from Th. 5 of § 3, No. 6.

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

Cited by

Corollary, § 4
Corollary, § 4
Proposition 7, § 4
Proposition 8, § 4
Proposition 4, § 6
Corollary 2, § 5
Corollary 5, § 5

Cites

Th. 5 of § 3, No. 6