Remark

Let $a$ be a real number $>0$. If one replaces $dx$ by the measure $a\cdot dx$, the convolution product of the functions $f$ and $g\in L^1(G)$ is replaced by $a(f*g)$. We have seen (II, p. 209, remark) that $\mathscr{F}(f)$ is replaced by $a\mathscr{F}(f)$. Hence $\mu_f$ is unchanged and $\nu$ is replaced by $a^{-1}\cdot \nu$. In particular, the measure $dx\otimes d\widehat{x}$ on $G\times \widehat{G}$ is independent of the choice of the Haar measure on G.