Lemma 1

Tag 09K5, permanent. Cite it as [Bourbaki, Tag 09K5].

Integration, VIII, p. 25, § 4. Convolution of measures and functions

Let $\mu$ be a measure on $G$ . Then $\chi$ is locally $(\mu \otimes \beta)$ -integrable, and the image of $\mu \otimes \beta$ under the homeomorphism $(s, x) \mapsto (s, s^{-1}x)$ of $G \times X$ onto $G \times X$ is $\chi \cdot (\mu \otimes \beta)$ .

We may suppose that $\mu \geqslant 0$ . Let $F \in \mathscr{K}_+(\mathbf{G} \times \mathbf{X})$ . Then

\iint F(s, s^{-1}x) d\mu(s) d\beta(x) = \int d\mu(s) \int F(s, s^{-1}x) d\beta(x)

= \int d\mu(s) \int F(s, x) d(\gamma(s^{-1})\beta)(x) = \int d\mu(s) \int F(s, x)\chi(s, x) d\beta(x).

Now, the function $(s, x) \mapsto F(s, x)\chi(s, x)$ has compact support and is $(\mu \otimes \beta)$ -measurable. By Ch. V, §8, No. 3, Prop. 7, the preceding equality proves that this function is $(\mu \otimes \beta)$ -integrable and that

\iint F(s, s^{-1}x) d\mu(s) d\beta(x) = \iint F(s, x)\chi(s, x) d\mu(s) d\beta(x).

This proves at the same time both assertions of Lemma 1.

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

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Cites

V, §8, No. 3, Prop. 7