Example

— 1) On the additive group $\mathbf{R}$, the Lebesgue measure $dx$ is a Haar measure (Ch. III, §1, No. 3, Example).

2) For every function $f \in \mathscr{K}(\mathbf{R}_+^*)$, we have (FRV, II, §1, formula (12))

for all $t > 0$; the measure $x^{-1} dx$ is thus a Haar measure on the multiplicative group $\mathbf{R}_+^*$.

3) Let us take for $G$ the torus $T = \mathbf{R}/\mathbf{Z}$. Let $\varphi$ be the canonical mapping of $\mathbf{R}$ onto $T$. For $f \in \mathscr{K}(T)$, the function $f \circ \varphi$ is continuous and periodic with period 1 on $\mathbf{R}$, and the integral

is independent of the choice of $a \in \mathbf{R}$; it is immediate that it is invariant under translation; it therefore defines a Haar measure on $T$. By transport of structure, one deduces from this that $I(f) = \int_a^{a+1} f(e^{2\pi i t}) dt$ is a Haar measure on the multiplicative group $U$ of complex numbers of absolute value 1 (GT, VIII, §2, No. 1).