Proposition 1

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

Let $\lambda$ be a measure on X, $\mu$ a measure on Y; if $g \in \mathscr{C}(X; \mathbf{C}),\ h \in \mathscr{C}(Y; \mathbf{C}),$ then

(g \cdot \lambda) \otimes (h \cdot \mu) = (g \otimes h) \cdot (\lambda \otimes \mu).

For every function $f \in \mathscr{K}(X \times Y; \mathbf{C})$ , we have, by virtue of formula (3) of No. 1,

\langle f, (g \otimes h) \cdot (\lambda \otimes \mu) \rangle = \int d\lambda(x) \int f(x, y) g(x) h(y) d\mu(y) \\ = \int g(x) d\lambda(x) \int f(x, y) h(y) d\mu(y),

which proves formula (6).

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

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