We consider for every $r\in \mathbb{R},$ two Borel functions $f_r:\mathbb{R}\to \overline{\mathbb{R}}$ and $h_r:\mathbb{R}\to \overline{\mathbb{R}}$ such that for every $r\in \mathbb{R},f_r\leq h_r$ and $\lambda(\{f_r\neq h_r\})=0,\lambda$ is the Lebesgue measure.
We suppose that for every $x\in \mathbb{R},$ the functions $r\to -f_r(x)$ and $r\to h_r(x)$ are upper semi-continuous.
Can we claim that for $\lambda$-almost every $x\in \mathbb{R},$ for all $r\in \mathbb{R},f_r(x)=h_r(x)$ (that is $\{x\in \mathbb{R},\exists r\in \mathbb{R},f_r(x)\neq h_r(x)\}$ is $\lambda$-negligeable)? Why?
I tried using Baire theorem to approximate by two monotone sequences of continuous functions $f^n_r,h^n_r$.
Any ideas?