Corollary 2

Tag 08V1, permanent. Cite it as [Bourbaki, Tag 08V1].

Integration, IV, p. 63, § 5. Measurable functions and sets

— The measurable sets in $X$ form a tribe (GT, IX, §6, No. 3).

For, if M and N are measurable then the functions

\varphi_{M \cup N} = \varphi_M + \varphi_N - \varphi_M \varphi_N \quad \text{and} \quad \varphi_{M \cap C_N} = \varphi_M - \varphi_M \varphi_N

are measurable by No. 3, Cors. 3 and 5 of Th. 1. If $(M_n)$ is a sequence of measurable sets and M is their union, then $\varphi_M = \sup_n \varphi_{M_n}$ is measurable by Cor. 1 of Th. 2, whence the corollary.

In particular, since the open sets are measurable:

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

Cited by

Theorem 5, § 5
Example, § 1
Definition 2, § 3

Cites

Th. 1
Cor. 1 of Th. 2
IX, §6, No. 3