Proposition 11

— Let $f, g$ be in $\mathscr{L}(G)$. Assume that one of the functions $f, g$ is continuous, and that one of the functions $f, g$ has compact support. Then $f$ and $g$ are convolvable. The formula (15) defines for all $x \in G$ a product $f * g$ that is continuous. If $f \in \mathscr{K}(G)$ and $g \in \mathscr{K}(G)$, then $f * g \in \mathscr{K}(G)$.

This follows from Props. 3 and 4 of No. 2.