Consider $L_p = L_p(\lambda^n)$ with the Lebesgue measure on $\mathbb{R}^n$ and $1 \leq p < \infty$.
Let $f_0(x) = |x|^{-\alpha}$ if $|x| < 1, f_{0}(x) = 0$ for $|x| \geq 1$.
Show that: $f_{0} \in L_p$ iff $p\alpha < n$
I'm having trouble understanding Lebesgue integration. Specifically I don't see where the n comes in. There are more similar questions to this one but if someone explained this one I think I could solve the rest.
Thanks
It should be known that $$ \int_{\mathbb{R}^N} f(x)\, d\mathcal{L}^N(x) = \int_{(0,+\infty) \times S^{N-1}} f(\rho \omega) \rho^{N-1} \, d\rho\, d\mathcal{H}^{N-1}(\omega), $$ where $\mathcal{L}^N$ is the $N$-dimensional Lebesgue measure and $\mathcal{H}^{N-1}$ is the Hausdorff measure on the sphere $S^{N-1}$. See also this link. If $f=f(|x|)$, then the result is a multiple of the 1D integral $$ \int_0^\infty f(\rho)\rho^{N-1}\, d\rho, $$ which, in your case...