SCHWEITZER 2012 Problem 8

For any function $f: \mathbb{R}^2\to \mathbb{R}$ consider the function $\Phi_f:\mathbb{R}^2\to [-\infty,\infty]$ for which $\Phi_f(x,y)=\limsup_{ z \to y} f(x,z)$ for any $(x,y) \in \mathbb{R}^2$ . [list=a] []Is it true that if $f$ is Lebesgue measurable then $\Phi_f$ is also Lebesgue measurable?[/] []Is it true that if $f$ is Borel measurable then $\Phi_f$ is also Borel measurable?[/] [/list]