Why $\frac{dx}{|x|}$ is a Haar measure on $\mathbb{R}\setminus\{0\}$

Question

This is in "A course in the abstract harmonic analysis by G.B. Follan on page 45"

Why $\frac{dx}{|x|}$ is a Haar measure on multiplicative group $\mathbb{R}\setminus\{0\}$

I've started as following

$\frac{dax}{|ax|}=\frac{adx}{|ax|}$

but I couldn't figure out the result

Any help will be greatly appreciated

Answer 1

$\int_{aE} \frac 1 {|x|} \, dx =\int_{E} \frac 1 {|x|} \, dx$ for all Borel sets $E$ in $\mathbb R\setminus \{0\}$ and all $a \neq 0$ (by an obvious change of variable). This is the definition of Haar measure.