I want to consider the function
$f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $
And I have shown that $\int_{[-1,1]}^{}\int_{[-1,1]}^{}f(x,y)d(y)d(x)=\int_{[-1,1]}^{}\int_{[-1,1]}^{}f(x,y)d(x)d(y)=0$
But f is not Lebesgue integrable.
How can this be happen?,since the above integrals are zero.