Fubini Thm to prove $\int _{[0,\infty)^n \setminus [0,1]^n} \frac{1}{\sum_1^n x_i^{a_i}} d\lambda <\infty \Leftrightarrow \sum_1^n\frac{1}{a_i} <1$

Question

This question have an answer which doesn't use Fibini-Tonelli here:

A problem in n-dimesnsional Lebesgue measure

The proposition states that $a_1,a_2,...,a_n >0$: $\int _{[0,\infty)^n \setminus [0,1]^n} \frac{1}{\sum_1^n x_i^{a_i}} d\lambda <\infty \Leftrightarrow \sum_1^n\frac{1}{a_i} <1$

I tried to work it out using Fubini Tonelli theorem, and by induction, by splitting the integral to 2 integral of $l,m$ dimensions such that $l+m=n$. I tried to work out how to split that $\sum_1 ^{n} \frac{1}{a_i}$ in order to apply the induction assumption. But I didn't succeed to find a way to do so.

I wish for a solution using Fubini theorem, I also hope it is not considered a duplicate, please let me know if so.