Theorem 1

Tag 08O8, permanent. Cite it as [Bourbaki, Tag 08O8].

Integration, chapter III, § 4. Products of measures

— Let X and Y be two locally compact spaces, $\lambda$ a measure on X, $\mu$ a measure on Y; there exists one and only one measure $\nu$ on $X \times Y$ such that, for every function $g \in \mathscr{K}(X; \mathbf{C})$ and every function $h \in \mathscr{K}(Y; \mathbf{C})$ ,

\int g(x)h(y)\, d\nu(x, y) = \left( \int g(x)\, d\lambda(x) \right) \left( \int h(y)\, d\mu(y) \right).

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

Cited by

Definition 1, § 4
Lemma 2, § 4
Proposition 2, § 4
Proposition 10, § 2
Remark, § 6
Proposition 9, § 6
Proposition 1, § 8
Proposition 10, § 8
Proposition 11, § 8
Proposition 12, § 8
Corollary, § 1
Remark, § 2
Proposition 10, § 2
Lemma 7, § 2
Proposition 1, § 2
Lemma 4, § 2
Proposition 1, § 2
Remark 1, § 4
Corollary 1, § 4

Cites

Nothing.