Proposition 6

*Assume $\chi$ a continuous multiplier and each function $\chi(s, \cdot)$ bounded. Let $\rho(s) = \sup_{x \in X} \chi(s^{-1}, x)$. Let $p$ and $q$ be two conjugate exponents ($1 \leq p < +\infty$). Let $\mu \in \mathscr{M}^{\rho^{1/q}}(G)$ and $f \in L^p(X, \beta)^2$ Then: (i) $\mu$ and $f$ are convolvable; (ii) the convolution product $\mu *^\beta f$ is given locally $\beta$-almost everywhere by the formula (3), and is equal locally $\beta$-almost everywhere to a function $g \in L^p(X, \beta)$ such that $\|g\|_p \leq \|\mu\|_{\rho^{1/q}} \|f\|_p$; (iii) $g$ is equal to the element $\gamma_\chi(\mu)f$ defined by the continuous representation $\gamma_\chi$ of $G$ in $L^p(X, \beta)$.

We have

by §2, No. 5, formula (5). On the other hand, the mapping $s \mapsto \gamma_\chi(s)f$ of $G$ into $L^p(X, \beta)$ is continuous (§2, No. 5, Prop. 9). Therefore this mapping is $\mu$-integrable. Let We have $\|g\|_p \leq (\int \rho^{1/q}(s) d|\mu|(s)) \|f\|_p$. Applying the preceding remarks to $|f|$, one sees that the mapping $s \mapsto \varepsilon_s * |f|$ of $G$ into $L^p(X, \beta)$ is $\mu$-integrable, therefore that, for every $h \in \mathscr{H}(X)$, the mapping $s \mapsto \langle h, \varepsilon_s * (|f| \cdot \beta) \rangle$ is $\mu$-integrable. Prop. 7 of §1, No. 5 then proves that $\mu$ and $f \cdot \beta$ are convolvable. Moreover, and this last integral is equal to $\langle h, \mu * (f \cdot \beta) \rangle$ by Prop. 7 of §1, No. 5. One therefore sees that $g$ is a convolution product of $\mu$ and $f$. This convolution product is given locally $\beta$-almost everywhere by (3), by Prop. 2 and the Remark that follows it.

2For a function $f$, the expression " $f \in L^p(X, \beta)$ " is an abuse of notation signifying that, depending on the context, the symbol $f$ is to be interpreted either as a function defined $\beta$-almost everywhere, or as the equivalence class of such a function for the relation of equality $\beta$-almost everywhere. Thus $f \in L^p$ can symbolize either $f \in \mathscr{L}^p$ or $\dot{f} \in L^p$.

By an abuse of notation, it is often one of the functions $g$ of the statement that is denoted $\mu *^\beta f$, which permits writing

If $X$ is countable at infinity, this style of notation is, moreover, entirely justified.