Proposition 2

— Let $\mu$ be a measure on G, f a locally $\beta$-integrable complex function on X. Assume that one of the following conditions is satisfied: (i) f and $\chi$ are continuous; (ii) G operates properly in X and f is zero on the complement of a countable union of compact sets; (iii) $\mu$ is carried by a countable union of compact sets. If $\mu$ and f are convolvable, then 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 one has, locally almost $\beta$-everywhere,

Let $h \in \mathscr{H}(X)$. Since $\mu$ and f are convolvable, the function $(s, x) \mapsto h(sx)f(x)$ is essentially $(\mu \otimes \beta)$-integrable. By Lemma 1, the function $(s, x) \mapsto h(x)f(s^{-1}x)\chi(s^{-1}, x)$ is essentially $(\mu \otimes \beta)$-integrable. Under hypothesis (i) or (ii) of the statement, one then deduces that this function is $(\mu \otimes \beta)$-integrable; for, in the first case it is continuous and one applies Prop. 3 of Ch. V, §1, No. 1, and in the second case it is zero outside a countable union of compact sets, and one applies Prop. 7, 2) of No. 2, loc. cit. By the Lebesgue–Fubini theorem,

the function $x \mapsto g(x) = \int f(s^{-1}x)\chi(s^{-1}, x)\, d\mu(s)$ being moreover locally $\beta$-integrable. One thus sees that

whence $g = \mu *_{\beta} f$.

Suppose now that $\mu$ is carried by the union S of a sequence of compact sets. The function

is essentially $(\mu \otimes \beta)$-integrable, and zero outside a countable union of compact sets, hence $(\mu \otimes \beta)$-integrable. Since $\mu = \varphi_S \cdot \mu$, the argument concludes as before.