The Scorza-Dragoni theorem is:
Suppose $f:\Omega \times \mathbf{R}^N \to R$ is measurable in its first variable and uniformly continuous in its second variable. Then for every $\epsilon>0$ there exists a compact set $K\subset \Omega$ such that $f|_{K \times \mathbf{R}^N}$ is continuous and $\text{meas}(\Omega \backslash K)<\epsilon$.
Apparently this result is also true if $f$ is only assumed to be continuous and not uniformly continuous in its second variable.
Where can I find a reference/proof of this?
I'm not 100% sure of this answer so any feedback would be appreciated. A basically equivalent form of what I have above is:
Suppose $f$ is measurable in its first variable and continuous in its second, and that $S \subset \mathbf{R}^N$ is compact. For every $\epsilon>0$ there is a compact $K\subset \Omega$ such that $|\Omega \backslash K|=\epsilon$ and $f|_{K\times S}$ is continuous.
To get $S=\mathbf{R}^N$, let $S_n\subset \mathbf{R}^N$ be the closed ball of radius $n$ and $\epsilon_n = \epsilon 2^{-n}$. Then there is a $K_n$ such that $f|_{K\times B_n}$ is continuous. Consider $K=\bigcap_n K_n$. Then $f$ is continuous on $K\times \mathbf{R}^N$ and $|\Omega \backslash K| \leq \sum_n |\Omega \backslash K_n| = \epsilon$.