Lusin Theorem.

Maybe we can avoid the case $C_0=\emptyset$ using the regualirity of Lebesgue measure: we can find a compact $K_{\varepsilon}$ contained in $A$ such that $m(A\setminus K_{\varepsilon})<\varepsilon /2$, then work with $K_{\varepsilon}$. The fact that an intersection of non empty compacts is non empty.