juanberrio8a

2022-06-27

I'm looking to show that $\frac{1}{|x{|}^{n}}$ is not integrable on the subset of ${\mathbb{R}}^{n}$ where $|x|\ge 1$. It's easy to do on $\mathbb{R}$, and I think I need to apply Fubini's theorem for the general case, but not sure how to do so in n-dimensional space. Thanks

Do you have a similar question?

Expert

Via some computation and Tonelli's theorem, we have for measurable $f:{\mathbb{R}}^{n}\to \left[0,\mathrm{\infty }\right]$ that
${\int }_{{\mathbb{R}}^{n}}f\left(x\right)\phantom{\rule{thinmathspace}{0ex}}dx={\int }_{{S}^{n-1}}{\int }_{0}^{\mathrm{\infty }}f\left(r\omega \right){r}^{n-1}\phantom{\rule{thinmathspace}{0ex}}dr\phantom{\rule{thinmathspace}{0ex}}dS\left(\omega \right),$
where $\phantom{\rule{thinmathspace}{0ex}}dS$ is "surface measure" on ${S}^{n-1}$. Now you can plug in $f\left(x\right)=\frac{1}{|x{|}^{n}}{\chi }_{\left\{x:|x|\ge 1\right\}}$ and compute it's integral.

Still Have Questions?

Free Math Solver