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On Wikipedia and also in my script it says that the definition of "integrable" is that a function $f$ is integrable if: $$ \int_\Omega \mid f \: \mid d\mu \lt \infty$$ Also in the condition for Fubinis Theorem to apply for the function f, is, that it has to be integrable (with resprect to the product measure). Now I've found several counterexpample for Fubini (Counterexample to Fubini?) but they always end in the one integral having the value $x$ and the other one $-x$, but $x$ is always $\lt \infty$.
My question is how do i see that a function like $\frac{x^2-y^2}{(x^2+y^2)^2}$ , with $\int_0^1\int_0^1 \frac{x^2-y^2}{(x^2+y^2)^2} dy\,dx$ beeing dependent on the order of integration, is non-integrable (it has to be non integrable since otherwise fubini would apply, or is my mistake already there?)

Any help is appreciated, thanks in advance!

1 Answers 1

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$$\int_0^1 |f(x,y)| dy = \int_0^x f(x,y) dy - \int_x^1 f(x,y)dy = \frac1{1+x^2} + \frac1x$$

Hence, by Tonelli's theorem, one has:

$$\int_{[0,1]^2} |f(x,y)|d(x,y) = \int_0^1 \left[ \int_0^1 |f(x,y)|dy \right]dx = \int_0^1 \left(\frac1{1+x^2} + \frac1x\right)dx = + \infty$$

So $f$ is not integrable on $[0,1]^2$.