Corollary

— Let $f \in L^2(G, \beta)$, $g \in L^2(G, \beta)$. Then $f$ and $\dot{g}$ are convolvable. One of the convolutions $f * \dot{g}$ belongs to $\overline{\mathscr{K}(G)}$ and its value at $e$ is $\int_G f(s)g(s) d\beta(s)$.

It suffices to take $p = q = 2$ in Prop. 15 and to apply (16).

We no longer assume $\beta$ to be left-invariant. Let $\rho$ be a lower semi-continuous finite function $> 0$ on $G$, such that $\rho(st) \leq \rho(s)\rho(t)$ for all $s, t$ in $G$. We denote by $L^\rho(G, \beta)$ the set of equivalence classes of the complex functions on $G$ that are integrable for $\rho \cdot \beta$. By the mapping $f \mapsto f \cdot \beta$, $L^\rho(G, \beta)$ may be identified with the set of elements of $\mathscr{M}^\rho(G)$ that have base $\beta$ (a set that is independent of the choice of $\beta$). If one sets

for $f \in L^\rho(G, \beta)$, this identification is compatible with the norms, thus $L^\rho(G, \beta)$ appears as a complete normed subalgebra of $\mathscr{M}^\rho(G)$. It is even a two-sided ideal of $\mathscr{M}^\rho(G)$ by Prop. 10 of §3, No. 2. (For $\rho = 1$, one recovers one of the assertions of No. 4.) In particular, $L^1(G, \beta)$ may be identified with a closed two-sided ideal of $\mathscr{M}^1(G)$.