Scholium

Tag 097N, permanent. Cite it as [Bourbaki, Tag 097N].

Integration, chapter V, § 8. Products of measures

— Let $f$ be a function defined on $T \times T'$ , with values in $\overline{\mathbf{R}}$ or in a Banach space, that is $\nu$ -measurable and $\nu$ -moderated. For the three integrals

\iint f(t, t')\, d\mu(t)\, d\mu'(t'), \quad \int d\mu(t) \int f(t, t')\, d\mu'(t'), \quad \int d\mu'(t') \int f(t, t')\, d\mu(t)

to exist and be equal, it is necessary and sufficient that one of the two numbers

\int^* d\mu(t) \int^* |f(t, t')|\, d\mu'(t'), \quad \int^* d\mu'(t') \int^* |f(t, t')|\, d\mu(t)

be finite.

This is an immediate consequence of Th. 1, Prop. 7 and the integrability criterion (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. 7
Th. 1