Show if function is Lebesgue-measurable I want to show if $f(x_1,x_2)=\dfrac1{1-x_1x_2}$ is Lebesgue-measurable or not on $[0,1)^2$. How do I start in this case, because the function is 2 variables? Normally, I would look if the set $\\{f>a\\}$ is in the $\sigma$-algebra $\forall$ $a$, as I have practised it before with Borel sets. I do not think it is useful here. i read on Internet that a.e.-continuous implies measurable? As this is homework, could please someone provide a hint?

Continuous functions are Borel measurables, hence Lebesgue measurable. Can you use that fact, or do you see how to prove it?