Theorem 1

Tag 096A, permanent. Cite it as [Bourbaki, Tag 096A].

Integration, chapter V, § 7. Integration with respect to an induced measure

— Let g be a function defined on X, with values in $\overline{\mathbf{R}}$ or in a Banach space. For g to be essentially $\mu_X$ *-integrable, it is necessary* and sufficient that $g$ be essentially $\mu$ -integrable in $X$ (§ 5, No. 3, Example), in which case

\int g \, d\mu_X = \int_X g \, d\mu.

This follows from Th. 2 of § 4.

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

Cited by

Remark, § 7
Scholium, § 7
Corollary 2, § 7
Corollary, § 1
Proposition 2, § 1
Corollary 1, § 2

Cites

Th. 2 of § 4
§ 5, No. 3