Our professor started a course in measure theory by stating the problems of Riemann integration. One of the problems he\she stated is the following double integration:
$\int_{0}^{1}\int_{0}^{1} \frac{x^2 – y^2}{(x^2 + y^2)^2} dy dx = \frac{-\pi}{4}$ but $\int_{0}^{1}\int_{0}^{1} \frac{x^2 – y^2}{(x^2 + y^2)^2} dx dy = \frac{\pi}{4}.$
My question is:
I have studied Lebesgue integration but still, until now it is not clear for me how Lebesgue integration solved the problem of changing the order of integration will change the value of integration? Is it by Fubini? if so, what was the solution?
Could anyone explain this to me, please?
Best Answer
If $f:[0,1]^2 \to \mathbb{R}$ is Lebesgue integrable, then we would have $\int_{[0,1]^2} |f| < \infty$ and Fubini's theorem would guarantee that the interated integrals are equal.
However, in this case, using polar coordinates, we have
$$\{(r,\theta): 0\leqslant r \leqslant 1, 0 \leqslant \theta \leqslant \pi/2\} \subset [0,1]^2,$$
and we see that
$$\int_{0}^{1}\int_{0}^{1} \frac{|x^2 - y^2|}{(x^2 + y^2)^2} \,dy\, dx > \int_0^{\pi/2}\int_0^1\frac{|r^2\cos^2 \theta - r^2\sin^2 \theta|}{(r^2)^2}\, r \, dr \, d\theta \\= \int_0^1 \frac{dr}{r}\int_0^{\pi/2}|\cos^2 \theta - \sin^2 \theta| \, d\theta = \infty$$