Corollary 3

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

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

Let $\theta$ be a complex measure on T; then

|\theta|_X = |\theta_X|.

Set $|\theta| = \mu$ and apply Cor. 2 on taking $g$ to be a complex function of absolute value 1 such that $\theta = g \cdot \mu$ (§ 5, No. 5, Cor. 3 of Th. 2); then $\theta_X = g_X \cdot \mu_X$ ; but $g_X$ is a function of absolute value 1, and the formula (6) follows from Prop. 2 of § 5, No. 2.

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

Cites

Prop. 2 of § 5, No. 2
§ 5, No. 5, Cor. 3 of Th. 2
Cor. 2