Corollary 2

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

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

— Let $(\mathbf{f}_n)$ be a sequence of integrable functions such that the series with general term $\mathbf{f}_n(x)$ converges almost everywhere; if there exists a function $g \geqslant 0$ such that $\int^* g d|\mu| < +\infty$ and such that, for every integer $n$ , $\left| \sum_{k=1}^n \mathbf{f}_k(x) \right| \leqslant g(x)$ almost everywhere, then the sum $\mathbf{f}(x)$ (defined almost everywhere) of the series with general term $\mathbf{f}_n(x)$ is integrable and

\int \mathbf{f} d\mu = \sum_{n=1}^\infty \int \mathbf{f}_n d\mu

('term-by-term integration of a series').

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

Cited by

Proposition 9, § 4

Cites

Nothing.