Proposition 6

Suppose that G is compact. The normalized Haar measure $dx$ belongs to $\mathscr{M}^1(G)$, and its Fourier transform is $\varphi_e$, the characteristic function of $\{e\}$.

Let $\chi \in \widehat{G}$. Since $\varepsilon_y*dx=dx$ for every $y\in G$, one has

by formula (11). If $\chi \not= 1$, there exists $y\in G$ such that $\langle \chi , y\rangle \not= 1$, hence $\mathscr{F}(dx)(\chi ) = 0$. If $\chi = 1$, then $\mathscr{F}(dx)(\chi ) =\int_Gdx= 1$ since the measure $dx$ is normalized.