So sorry I'm very new to the Theory of Measure and Integration. I failed the task and I don't know how it can be solved.
I've started using Fubini Theorem for the sigma-finite measure, but it is the wrong way. Of course I don't know how it should be
$\int_{[0,1]^{2}}f(x,y)d\mu$ = $\int_{[0,1]} \left ( \int_{[0,1]} f(x,y)d\mu_{Y} \right )d\mu_{X}$
where $f$ is a simple integrated function, and $\mu_{{X}\oplus{Y}}$ is a product Lebesgue' measure