Proposition 12

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

Integration, chapter V, § 8. Products of measures

— Let X (resp. X') be a locally compact subspace of T (resp. T'). Then, the induced measure $(\mu \otimes \mu')_{X \times X'}$ on the locally compact subspace $X \times X'$ of $T \times T'$ is equal to the product $\mu_X \otimes \mu'_{X'}$ of the measures induced on X and X' by $\mu$ and $\mu'$ , respectively.

For, if $f \in \mathscr{K}(X)$ and $f' \in \mathscr{K}(X')$ , then

\iint_{X \times X'} f(t)f'(t') d\mu(t) d\mu'(t') = \left( \int_X f(t) d\mu(t) \right) \left( \int_{X'} f'(t') d\mu'(t') \right)

by Cor. 2 of Prop. 8, which proves, by the definition of induced measures (Ch. IV, §5, No. 7) that

(\mu \otimes \mu')_{X \times X'} = \mu_X \otimes \mu'_{X'}

(Ch. III, §4, No. 1, Th. 1).

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

Cited by

Nothing.

Cites

III, §4, No. 1, Th. 1
IV, §5, No. 7
Cor. 2 of Prop. 8