Suppose that $f: U\times \mathbb R^m\to \mathbb R$ is a measurable function where $U\subseteq \mathbb R^n$ is a measurable set. Also let $K\subseteq \mathbb R^m$ is a compact set.
Then prove that the set $$\Phi(x)=\sup\{ |f(x,u)-f(x,v)|:u,v\in K, ||u-v||<1\} ,\text{ for all }x\in U$$ is a measurable function.
Everything is taken as Lebesgue measure.
I know that if we take supremum over the countable number of measurable functions then the function is measurable.
Any help is appreciated. Thank you