Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue measurable. Prove for any $\epsilon > 0$ there exists $g\in C([0,1])$ st.
$$|\{x: f(x) \neq g(x)\}| < \epsilon$$

This seems to be a standard result, but I couldn't find a proof. Maybe someone can give a reference? Thank you