Let $E$ be a subset of $X\times Y$ and every section of $E$ is measurable then is $E$ measurable in the product sigma algebra of $X\times Y$??

Question

Definition of section is $E_x=\{y\in Y:(x,y)\in E\}$ is a subset of $Y$ for all $x\in X$..and similarly define $E_y=\{x\in X:(x,y)\in E\}$ is a subset of $X$... now for two measurable spaces $X,Y$ if $E_x$ is measurable in $Y$ for all $x\in X$ and $E_y$ is measurable in $X$ for all $y\in Y$ for a set $E$ is a subset of $X\times Y$ then is $E$ measurable in product sigma algebra of $X\times Y$?? any hint will be appreciated..

Answer 1

Pick $ X=Y= [0;1]$ with the Borel-sigma algebra and let $A\subseteq [0;1]$ be not Borel-measurable. Then consider $E=\{ (a,a)\in [0;1]^2 \ : \ a\in A \}$. All the sections are singeltons or empty and hence measurable. We are left to show that $E$ is not measurable. For this we note that $f: [0;1] \rightarrow [0;1]\times [0;1], x \mapsto (x,x)$ is continuous and hence measurable (here we use that we put the Borel-sigma algebra). Thus, if $E$ was measurable, then $A= f^{-1}(E)$ would be measurable too.