Finding $E in mathcal A otimes mathcal B$ such that $E neq E^y times E_x,$ for some $x in X, y in Y.$

Let $(X, mathcal A, mu)$ and $(Y, mathcal B, nu)$ be two measure spaces. What has been stated in my book is that $mathcal A times mathcal B$ may not necessarily be a $sigma$-algebra of subsets of the product space $X times Y$ but it’s a semi-algebra of subsets of $X times Y.$ Let $mathcal A otimes mathcal B$ be the $sigma$-algebra of subsets of $X times Y$ generated by $mathcal A times mathcal B.$ Then the following theorem holds $:$

Theorem $:$ For every $E in mathcal A otimes mathcal B,$ for every $x in X$ and for every $y in Y$ $$E_x in mathcal B text {and} E^y in mathcal A$$
where $E_x$ and $E^y$ are the $x$-section and $y$-section of $E$ respectively defined by begin{align*} E_x : & = left {y in Y | (x,y) in E right } \ E^y : & = left {x in X | (x,y) in E right } end{align*}

Does the above theorem imply that $E = E^y times E_x $? I don’t think so. For otherwise $mathcal A otimes mathcal B = mathcal A times mathcal B,$ which is not necessarily true. Can anybody help me finding one such example where $E in mathcal A otimes mathcal B$ but $E neq E^y times E_x $?

Thanks in advance.