For what $\alpha \in \mathbb R$ is $\int_{\{\mathbf{|x|} \in \mathbb{R}^d : \mathbf{|x|} \le 1\}} \mathbf{|x|^{- \alpha}} dm(\mathbf{x})$ finite?

Question

I'm studying the Lebesgue integral and have run into trouble with this problem. The integral I'm trying to work with is:

$\int_{\{\mathbf{|x|} \in \mathbb{R}^d : \mathbf{|x|} \le 1\}} \mathbf{|x|^{- \alpha}} dm(\mathbf{x})$

I had tried this and managed to do it for $d =1,2$ by drawing and visualizing it and feel the answer is $\alpha \lt d$. I feel the proof might be inductive but am unable to prove it that way. Another approach, hinted at by a friend I'm working with is to perhaps convert the problem to generalized polar coordinates but I am unable to do it that way. Is there some other, more simple way to do it that I might not be seeing?

Thanks.

Answer 1

This well explained in the following theorem in Folland's Real Analysis