Let $f\colon \mathbb{R}^2\rightarrow \mathbb{R},\ f(x,y)= \begin{cases} \frac{\mathrm{sign}(xy)}{x^2+y^2}, \ (x,y)\ne 0\\ 0, \ (x,y)=0 \end{cases}$
Both iterated integrals are zero and therefore equal, but $f(x,y)$ is not Lebesgue integrable on $\mathbb{R}^2$.
That $f(x,y)$ is not L-integrable was shown the following way:
$\int_{\mathbb{R}^2}|f|\,\mathrm{d}(x,y)=\int_{\mathbb{R}}\int_{\mathbb{R}}|f_y(x)|\,\mathrm{d}x\mathrm{d}y=\infty$.
From this the conjecture was concluded.
But how can this be possible? We have that $\left | \int_X f\,\mathrm{d}\mu\right |\le \int_X |f|\,\mathrm{d}\mu$ which means that it is still possible that the non absolute value of the function is L-integrable.
What do I miss?
The following should be known: if $f$ is measurable, then:
$f$ is integrable $ \iff |f|$ is integrable.