Let $(X,\mathcal{A})$, $(Y,\mathcal{B})$ be measurable spaces and let $f:(X\times Y,\mathcal{A}\otimes \mathcal{B})\to(\mathbb R,\mathcal B(\mathbb R))$ be measurable map. Consider the $\sigma$-algebra $\sigma(f_x:x\in X)$ in $Y$ generated by the maps $f_x$ for $x\in X$. Then $\sigma(f_x:x\in X)\subset \mathcal{B}$, and so $$ \mathcal{A}\otimes \sigma(f_x:x\in X) \subset \mathcal{A}\otimes \mathcal{B}.$$
Question. Can I find such an $f$ which is not $\mathcal{A}\otimes \sigma(f_x:x\in X)$ measurable?
Here we can find an example of a map $f$ which is not $\mathcal{A}\otimes \mathcal{B}$ measurable despite having all its $x$ and $y$ sections measurable.
Thanks a lot for your help.
