Counterexample to Fubini? I am trying to come up with a measurable function on $[0,1]^2$ which is not integrable, but such that the iterated integrals are defined and unequal.
Any help would be appreciated.

$$
\int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \,dy\,dx \ne \int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} \,dx\,dy
$$
Obviously either of these is $-1$ times the other and if this function were absolutely integrable, then they would be equal, so their value would be $0$.  But one is $\pi/2$ and the other is $-\pi/2$, as may be checked by freshman calculus methods.
