Example

— 1) Lebesgue measure on $\mathbf{R}^n$ is a Haar measure of the additive group $\mathbf{R}^n$.

2) The mapping $(r, u) \mapsto ru$ is an isomorphism of $\mathbf{R}_+^* \times \mathbf{U}$ onto $\mathbf{C}^*$ (GT, VIII, §1, No. 3). If $\mathbf{C}^*$ is identified with $\mathbf{R}_+^* \times \mathbf{U}$ by means of this isomorphism, and if $du$ denotes a Haar measure on $\mathbf{U}$, then $r^{-1} dr du$ is a Haar measure on $\mathbf{C}^*$ by Example 2 of No. 2. On the other hand, the bijection $\theta \mapsto e^{2i\pi \theta}$ of $[0, 1[$ onto $\mathbf{U}$ transforms the Lebesgue measure $d\theta$ on $[0, 1[$ into a Haar measure on $\mathbf{U}$ by Example 3 of No. 2. It follows that if $f \in \mathscr{H}(\mathbf{C}^*)$, the integral

defines a Haar measure on $\mathbf{C}^*$.