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