Compactly supported convolution on a group

Let $G$ be an LCH group with a fixed left Haar measure $\mu$ (Left Haar integral and left Haar measure) and let $f,g\in C_c(G)$ be complex-valued of compact support (Complex Haar L^p spaces and compactly supported functions). Their convolution $f\ast g$ is the function $(f\ast g)(x):={\int }_{G}f(y)g(y^{-1}x)d\mu (y)(x\in G),$ with the displayed order of the factors and of the product $y^{-1}x$. The integral is taken over $G$ against the left Haar measure and is written with the same symbol $\ast$ in every later occurrence.

• Well-definedness for each $x$. For fixed $x$, the function $y\mapsto f(y)g(y^{-1}x)$ is continuous as a product of continuous functions of $y$, and it vanishes unless $y\in \mathrm{supp}f$; hence it is continuous with compact support and its integral is a finite complex number, because $\mu$ is finite on compact sets. The value $(f\ast g)(x)$ is thus defined for every $x\in G$ and no integrability of $f$ or $g$ beyond compact support is used.
• Order convention. The product is $g(y^{-1}x)$, not $g(x{y}^{-1})$: this is the order under which convolution makes $L^1(G)$ associative for a left Haar measure, as proved on the next item. Reversing the order silently would give the convolution suited to a right Haar measure.
• Noncommutativity. No symmetry of $f\ast g$ under $f\leftrightarrow g$ is asserted. For abelian groups the two orders agree, and only there is convolution written without concern for the side.