Theorem 1

Tag 094Y, permanent. Cite it as [Bourbaki, Tag 094Y].

Integration, chapter V, § 6. Images of a measure

— Let f be a function defined on X with values in $\overline{\mathbf{R}}$ or in a Banach space F. For f to be essentially $\nu$ -integrable, it is necessary and sufficient that $f \circ \pi$ be essentially $\mu$ -integrable, in which case

\int f(x)\, d\nu(x) = \int f(\pi(t))\, d\mu(t).

Suppose, moreover, that $\pi$ is continuous and proper. Then, for f to be $\nu$ -integrable, it is necessary and sufficient that $f \circ \pi$ be $\mu$ -integrable.

It suffices to apply Th. 2 of §4, No. 4.

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

Cited by

Corollary, § 6
Proposition 4, § 6
Proposition 5, § 6
Proposition 7, § 6
Proposition 9, § 6
Proposition 2, § 1
Theorem 1, § 3
Theorem 4, § 3
Proposition 2, § 1

Cites

Th. 2 of §4, No. 4