Lemma 3

Tag 0970, permanent. Cite it as [Bourbaki, Tag 0970].

Integration, chapter V, § 8. Products of measures

Let $f$ be a function defined on $X$ , with values in $\overline{\mathbf{R}}$ or in a Banach space. For $f$ to be $\lambda'_t$ -integrable, it is necessary and sufficient that $f_t$ be $\mu'$ -integrable, in which case

\int f_t \, d\mu' = \int f \, d\lambda'_t .

This follows from Th. 2 of §4, No. 4, on taking into account the fact that $t' \mapsto (t, t')$ is continuous and proper.

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

Cited by

Nothing.

Cites

Th. 2 of §4, No. 4