Past exam of the integration course of the University of Oxford / 2015 / 3 / a

Fix $y\ne 0$. The function $x\mapsto f(x,y)$ is continuous on $[-1,1]$, because its denominator is bounded away from zero, and it is an odd function of $x$. Its integral over this symmetric interval is therefore zero. At $y=0$ the entire section is zero, including the specified value at the origin. Thus the inner $x$ integral exists for every $y$ and is identically zero. The same reasoning with $x$ and $y$ exchanged givesTo test Lebesgue integrability in two dimensions, examine the absolute value near the origin. For $0<\epsilon <1$, the annulus $\epsilon <\sqrt{x^2+y^2}<1$ lies inside the square. In polar coordinates,The change of variables formula on this annulus, where the integrand has no singularity, givesThe lower bound tends to infinity as $\epsilon \downarrow 0$. HenceMoreover reflection in either coordinate axis exchanges the positive part and negative part while preserving Lebesgue measure. Both parts therefore have infinite integral, so a signed double Lebesgue integral would involve the undefined expression $\infty -\infty$.

There is no contradiction with Fubini's theorem: its absolute integrability hypothesis is missing. This is an example of finite iterated integrals without joint integrability, with every individual one-dimensional section integrable despite the two-dimensional singularity.