Every section of a measurable set is measurable? (in the product sigma-algebra)
Solution 1:
Since the map $u_{x}:Y\to X\times Y$ given by $u_x(y)=(x,y)$ for a given $x$ is $\mathcal{N}$-$(\mathcal{M}\otimes\mathcal{N})$-measurable and $E_x=u_x^{-1}(E)$ it follows directly that $E_x\in\mathcal{N}$ for any $x$.
To convince yourself that $u_x$ is indeed measurable, just note that $$ u_x^{-1}(A\times B)= \begin{cases} B,\quad &\text{if }x\in A,\\ \varnothing, &\text{if } x\notin A, \end{cases} $$ which is in $\mathcal{N}$ for any $A\in\mathcal{M}$ and $B\in\mathcal{N}$.