We have two measurable spaces $X$ and $Y$ (with measures $\mu_X$ and $\mu_Y$) and consider their product measurable space $Z=X \times Y$. Furthermore I have a measurable (null-)set $M \subseteq Z$. For every $x \in X$ I want to consider the set $M_x:=\{y \in Y \ | \ (x,y) \in M\}$ or written as projection $M_x=\pi_Y|_{(x,Y)}(M)$. The question is now for given $\varepsilon >0$:
Is the set $S=\{x \in X \ | \ \mu_Y(M_x)>\varepsilon\} \subseteq X$ measurable?
In words, is the set of x, which fiber have "big" measure in $Y$, measurable.
It is clear, that if this is the case, then $S$ is a nullset too (if $M$ is a nullset). The only problem right now, is to see it is indeed measurable.