No Tonelli&Fubini contradiction

Question

I was trying to solve the following question:
Let $f(x,y)=\cases{1/x^2: x>y\ge0\\ -1/y^2: y>x\ge0\\ 0: x=y}$
Show that $\intop_0^1dx\intop^1_0f(x,y)dy\ne\intop_0^1dy\intop_0^1f(x,y)dx$
I did show that, but another question was asked is to explain why we don't get a contradiction according to Fubini and Tonellis theorems. I wasnt able to explain that.
Can someone please give me a hand with that?
I think that it has something to do with $\sigma-finite$ or maybe that $f$ is not measurable, but I dont think this is the case here..

Answer 1

Tonelli's theorem is about non-negative functions. $f$ attains positive and negative values, so it is not one of the functions Tonelli's theorem is concerned with, hence the non-equality of the iterated integrals doesn't contradict Tonelli's theorem.

Fubini's theorem is about integrable functions. $f$ is not integrable, hence the iterated integrals being different doesn't contradict Fubini's theorem either.