Corollary

Tag 0918, permanent. Cite it as [Bourbaki, Tag 0918].

Integration, chapter V, § 3. Integration of positive measures

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

\int^\bullet d\mu(t) \int^\bullet |\mathbf{f}(x)|\, d\lambda_t(x) < +\infty.

This follows at once from Prop. 5 and the criterion for integrability (Ch. IV, §5, No. 6, Th. 5).

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

Cited by

Nothing.

Cites

IV, §5, No. 6, Th. 5
Prop. 5