Definition 1

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

Integration, chapter III, § 4. Products of measures

Given two measures $\lambda, \mu$ defined, respectively, on two locally compact spaces $X, Y$ , the product measure of $\lambda$ by $\mu$ is defined to be the unique measure $\nu$ on $X \times Y$ satisfying the relation (1) for every function $g \in \mathscr{K}(X; \mathbf{C})$ and every function $h \in \mathscr{K}(Y; \mathbf{C})$ .

In the proof of Th. 1, the roles of the spaces X and Y may be interchanged; canonically identifying $Y \times X$ and $X \times Y$ , we thus define on $X \times Y$ a measure

f \mapsto \int d\lambda(x) \int f(x, y) d\mu(y)

that again satisfies condition (1). We have thus proved the following theorem:

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

Cited by

Nothing.

Cites

Th. 1