Corollary 2

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

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

Let g be a complex function defined on T and locally $\mu$ -integrable; the restriction $g_X$ of g to X is then locally $\mu_X$ -integrable, and

(g \cdot \mu)_X = g_X \cdot \mu_X.

This follows at once from Th. 1, applied to the functions $fg_X$ ( $f \in \mathscr{K}(X; \mathbf{C})$ ), and the definition of the measure induced on X by a complex measure (Ch. IV, § 5, No. 7).

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

Cited by

Corollary 3, § 7

Cites

IV, § 5, No. 7
Th. 1