Remark

The Fourier transform on the space $L^1(G)$ depends on the choice of the Haar measure $dx$, unlike the Fourier transform on $\mathscr{M}^1(G)$. If one replaces $dx$ by the measure $a\cdot dx$ (with $a >0$), then for every integrable function $f$ on G, the Fourier transform of $f$ is $a\widehat{f}$, where $\widehat{f}$ is the Fourier transform defined relative to the measure $dx$.

Consider the stellar algebra Stell(G) of the group G (def. 9 of I, p. 125), and identify $L^1(G)$ with a dense subalgebra of Stell(G) (prop. 22 of I, p. 126).