So I'm trying to understand how to calculate ball integrals in measure theoretic way:
$\int_{\mathbb{R}^n \setminus B(0,r)} \frac{1}{|x|^{\alpha}}dx$
$=\int_{\mathbb{R}^n} \frac{1}{|x|^{\alpha}} \chi_{\mathbb{R}^n \setminus B(0,r)} (x)dx$
$=r^n\int_{\mathbb{R}^n} \frac{1}{|rx|^{\alpha}} \chi_{\mathbb{R}^n \setminus B(0,r)} (rx)dx$ $=...$
as written in this, page 3.
First two lines are clear, but I don't understand what happens in the third line. $r^n$ comes from somewhere, one adds $r$ inside $\chi$ and then there's $r^{\alpha}$ added to the integrand. What, why?
The linked paper also has this short theorem or lemma on top of p. 3, which gives a formula for
$$\int_{\mathbb{R}^n} f(|x|)dx=\omega_{n-1} \int_0^{\infty} f(r)r^{n-1} dr$$
I wonder if this can be found elsewhere?