Corollary 1

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

Integration, V, p. 76, § 7. Integration with respect to an induced measure

For a subset B of X to be essentially $\mu_X$ -integrable, it is necessary and sufficient that it be essentially $\mu$ -integrable, in which case $\mu_X(B) = \mu(B)$ .

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

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