Let $g: \mathbb{R}_{\geq 0} \to \mathbb{R}$ be a function and let $f: \mathbb{R}^n \to \mathbb{R}$ be $f(x)=g(|x|)$.
Show that $f$ is integrable in $\mathbb{R}^n$ iff $r \mapsto r^{n-1} g(r)$ is integrable in $\mathbb{R}_{\geq 0}$.
Can someone help me with this?
There is nothing to prove actually if $n=1.$ So we assume $n\ge 2.$ First of all note that every $0\neq x\in\mathbb{R}^n$ can be uniquely written as $x=r\omega$ where $r>0$ and $\omega\in S^{n-1}.$ The rest is essentially a change of variable. Note that $$\int_{\mathbb{R}^n} f(x)dx=\int g(|x|)dx=\int_{S^n} \int_{0}^{\infty}g(r)r^{n-1}drd\omega.$$
Now note that $\int_{S^n}d\omega=C_n$ is just the surface measure of sphere which is a constant (depending upon the dimension). The Fubini’s theorem therefore gives you the desired the conclusion.
