Corollary

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

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

Let $(A_n)$ be a sequence of integrable sets such that $\sum_{n=1}^\infty |\mu|(A_n) < +\infty$ ; the union $A = \bigcup_n A_n$ is integrable, and

|\mu| \left( \bigcup_n A_n \right) \leq \sum_{n=1}^\infty |\mu|(A_n).

For, $\varphi_A = \sup_n \varphi_{A_n}$ and

|\mu|^*(A) \leq \sum_{n=1}^\infty |\mu|^*(A_n) = \sum_{n=1}^\infty |\mu|(A_n) < +\infty

(§1, No. 4, Prop. 18); therefore A is integrable (§3, No. 6, Cor. 2 of Th. 5) and, since $|\mu|(A) = |\mu|^*(A)$ , we indeed have (17).

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

Cited by

Proposition 3, § 5
Theorem 2 (Egoroff), § 5

Cites

§1, No. 4, Prop. 18
§3, No. 6, Cor. 2 of Th. 5