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

Question

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.

Answer 1

Take both spaces to be the real line with Borel sigma algebra. Consider the open unit disk $E =\{(x,y): x^{2}+y^{2} <1\}$. Then $E_0=[-1,1]=E^{0}$ and $E_0 \times E^{0}=[-1,1] \times [-1,1] \neq E$.

Answer 2

Here's a classic example. Let $\cal{A}=\cal{B}$ be the Lebesgue measurable subsets of $\Bbb R$. Then $\Delta=\{(x,x):x\in\Bbb R\}$ is an element of $\cal{A}\otimes\cal{B}$ but not of $\cal{A}\times\cal{B}$.

In this case $\Delta_x=\Delta^x=\{x\}$ for all $x\in\Bbb R$.