Let $\|x\|= \biggl( \sum_{k=1}^n |x_k|^p \biggr)^{\!1/p\;}$ for a given $p>1$ and $f:\mathbb{R}^n\to [0,\infty)$ of the form $f(x)=g(\|x\|)$ for a given $g:[0,\infty)\to[0,\infty)$
Prove that if $g$ is integrable then $f$ is integrable and $\int_{\mathbb{R}^n} f =nV_1\int_0^\infty g(r)r^{n-1} \, dr$ where $V_a$ is the volume of $B_a=\{x;\|x\|\leq a\}$.
My work:
So I started by taking $g$ to be $\mathbb{1}_{[0,a]}$ for some $a>0$. In this case $f=\mathbb{1}_{B_a}$ and if $g$ is integrable then $f$ is too, then I took $g$ to be a step function which is a linear combination of indicator functions so again if $g$ is integrable so is $f$.
Now because every integrable function is sandwiched between two step functions with integrals that differ by $\epsilon$ we conclude that if $g$ is integrable $f$ is as well. (Not too sure about this part)
Now to prove the integral equality above, I checked it on $n=2$ and $p=2$ and it was true but I am pretty lost on how to do an induction here..Any ideas?