This is a question from Folland's Book, Suppose $$f(x,y)=(1-xy)^{-a}$$ and $a>0$. Check if the following integrals exist and if they are equal:
$$\int_{[0,1]\times[0,1]}f d(x,y),\int_0^1\int_0^1f(x,y)dxdy,\int_0^1\int_0^1f(x,y)dydx$$
I know it has to deal with Fubini Tonelli's Theorem and this is in the context of Lebesgue Integral.
I know that $$\int_{[0,1]\times[0,1]}fd(x,y)=\sum_{k\ge1}\int a_k\chi_{S_k\cap [0,1]\times[0,1]}d(x,y)$$ Since the function is non negative. It is also continuous except at $(1,1)$, so measurable.
If I could show that the above integral is finite, then I would be done. But no idea how to show the above is finite.
Best Answer
Changing coordinates to polar coordinates around the corner $(1,1)$ shows that the double integral converges if and only if $\int\limits_0r^{−a}\,r\mathrm dr$ does, that is, exactly for $a<2$.