Suppose $f: \mathbb R^n \to \mathbb R$ is a measurable function such that $|f(x)| \leq g(x)$, where $g(x) = c|x|^{-p} \chi_{B(0,1)}(x)$for some $c > 0, p <n$. Prove $f$ is integrable.
This is from Bass exercise 11.21.
Let $\tilde x=(x_1,...,x_{n-1}), x = (\tilde x, x_n)$, and $f^{x_n}(\tilde x) = f(x) : \mathbb R^{n-1} \to \mathbb R$.
If $p \leq 0$ then $g$ is integrable, so that $f$ also is, and we are done. So assume $p > 0$.
We are suggested to proceed by induction on $n$. The case for $n=1$ can be solved. Now assume the case holds for $n-1$, and let $\epsilon > 0$ be so small that $p+\epsilon$ is still smaller than $n$. Then $p-1 + \epsilon < n-1$, and we can show that $|f^{x_n}(\tilde x)| \leq c|\tilde x|^{-p+1-\epsilon} \chi_{\tilde B(0,1)}(\tilde x) |x|^{-1+\epsilon}$. A person suggested to me that I should somehow use Dominated Convergence theorem and the fact that $|x|^{-1+\epsilon}$ is integrable, but I am not sure how to proceed as such. Certainly, we may bound $g$ by $c|x|^p \chi_{[0,1]^n}(x)$, but I am not sure what to do here. It seems like I have to use Tonell-Fubini somewhere...
I apologizw for mixing up the precise problem statement. The main problem is now correct; if there is any error in signs please construe as close you think it should be. Thank you.
Of course, this could be done in polar coordinates but I am specifically asking for one using indunction.