Corollary 1

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

Integration, chapter V, § 8. Products of measures

*With notations as in Prop. 8,

\iint^* f(t)f'(t')\,d\mu(t)\,d\mu'(t') = \left( \int^* f(t)\,d\mu(t) \right) \left( \int^* f'(t')\,d\mu'(t') \right)

except possibly when one of the factors of the second member is equal to 0 and the other is equal to $+\infty$ .

When the two factors of the second member are finite, the functions $f$ and $f'$ are moderated (§ 1, No. 2, Prop. 7), therefore the function $f \otimes f'$ is moderated (Cor. 2 of Prop. 5); the above equality therefore reduces to the formula (10) (§ 1, No. 2, Prop. 7). When one of the factors of the second member has the value $+\infty$ and the other is not zero, then the second member has the value $+\infty$ , and the above equality follows from Prop. 6.

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

Cited by

Nothing.

Cites

§ 1, No. 2, Prop. 7
Cor. 2 of Prop. 5
Prop. 6
Prop. 8