Let $f(x) = |x|^{-a}$ if $|x| \leq 1$ and $0$ otherwise. Prove that $f$ is integrable on $\mathbb{R^d}$ if and only if $a$ < $d$.
going to need some help with this one, this Lebesgue integration stuff is so hard! If anyone can lend me some insight that'd be awesome.
So i'm thinking I'll have to do something with expressing f as the limit of a series of simple functions? I have so much experience calculating Riemann integrals, but haven't the slightest clue on how to work with Lebesgue integrals, so my first instinct is to go back to how we constructed them, but I can't help but get the feeling that this is unnecessary.
Hint: Note that
$$\int_{|x|<1} |x|^{-a}\,dm_d(x) = \sum_{n=0}^{\infty}\int_{2^{-n-1}<|x|<2^{-n}} |x|^{-a}\,dm_d(x) $$
You can make the simplest estimates possible on the terms of the sum to solve the problem.