I am trying to verify that the measure $\frac{1}{|x|}dx$ is a Haar measure on $\mathbb{R}\backslash \{0\}$.
For every open interval $(a_{n},b_{n})\subset \mathbb{R}$ not containing $0$, I have that the measure of $(a_{n},b_{n})$ is given by $$\int_{a_{n}}^{b_{n}}\frac{1}{|x|}dx = \ln(b) - \ln(a)$$
and thus the measure $x(a_{n},b_{n})$ is
$$\int_{xa_{n}}^{xb_{n}}\frac{1}{|x|}dx = \ln(xb) - \ln(xa) = \ln(b) - \ln(a)$$
Since the Borel subsets of $\mathbb{R}\backslash \{0\}$ are generated by such open intervals, am I done? Or do I still have to consider arbitrary Borel sets?
It seems that the following approach works to extend properties to $\sigma$-algebras.
First, prove that the property holds for all the generators of the $\sigma$-algebra$.
Second, prove that the collection of elements for which the property holds is a $\sigma$-algebra.