Proposition 1

Let $\mu$ be a measure on $\mathbf{G}$, $f$ a locally $\beta$-integrable complex function on $\mathbf{X}$. Suppose that the function $s \mapsto f(s^{-1}x)\chi(s^{-1}, x)$ is essentially $\mu$-integrable except for a locally $\beta$-negligible set of values of $x$, and that the function $x \mapsto \int |f(s^{-1}x)|\chi(s^{-1}, x) d|\mu|(s)$, defined locally almost everywhere for $\beta$, is locally $\beta$-integrable. Then $\mu$ and $f$ are convolvable.

We may assume that $f \geqslant 0$ and $\mu \geqslant 0$. Let $h \in \mathscr{K}_+(\mathbf{X})$. We are to prove that the function $(s, x) \mapsto h(sx)$ is essentially integrable for $\mu \otimes (f \cdot \beta) = (1 \otimes f) \cdot (\mu \otimes \beta)$ (Ch. V, §8, No. 5, Prop. 10), that is, that $\iint^\bullet h(sx)f(x)\varphi_K(s) d\mu(s) d\beta(x) < +\infty$ (Ch. V, §5, No. 3, Prop. 3); it will clearly suffice to prove that there exists an $a > 0$ such that for every compact subset $K$ of $\mathbf{G}$,

By Lemma 1, Now, the function $(s, x) \mapsto h(x)f(s^{-1}x)\varphi_K(s)\chi(s^{-1}, x)$ is $(\mu \otimes \beta)$-measurable (Lemma 1) and has compact support. The preceding expression is therefore equal (Ch. V, §8, No. 3, Prop. 7) to

where S denotes the support of h. Whence the proposition.