Lebesgue measure by closed sets

Question

Consider this definition: we say $E$ is Lebesgue measurable if for any $\epsilon>0$ there is a closed set $F$ such $m_{\star}(E-F)<\epsilon$. Show an open set is Lebesgue measurable (without assuming $E^{c}$ is measurable).

Answer 1

Here is a possible idea or hints:

1. Show that if each of the sets $E_n$ ($n \in \mathbf{N}$) is Lebesgue measurable by closed sets (LMC) then $\bigcup\limits_{n \in \mathbf{N}} E_n$ is also LMC.
2. Using the previous point, reduce to show the case where the open set $E$ has a compact closure.
3. Consider the set $E_r$ of points whose distance to the complement of $E$ is $\geq r.$
4. Show each $E_r$ is closed. Show finally $m_*(E\setminus E_r) \to 0$ as $r \downarrow 0.$