Proposition 8

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

Integration, IV, p. 42, § 4. Integrable functions and sets

Let $(A_n)$ be an increasing sequence of integrable sets; for the union $A = \bigcup_n A_n$ to be integrable, it is necessary and sufficient that $\sup_n |\mu|(A_n) < +\infty$ , in which case,

\mu(A) = \lim_{n \to \infty} \mu(A_n).

For, the $\varphi_{A_n}$ form an increasing sequence of integrable functions, and $\varphi_A = \sup_n \varphi_{A_n}$ ; thus, the proposition follows from Prop. 4 of No. 3.

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

Cited by

Corollary 2, § 4

Cites

Prop. 4