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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.

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  • $\begingroup$ 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 $a\lt2$. $\endgroup$ – Did Aug 28 '13 at 14:01
  • $\begingroup$ @Did Thanks. I did not even think from that perspective, changing coordinates. :( $\endgroup$ – user92039 Aug 28 '13 at 14:04
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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 $a<2$.

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