Relationship between the Lebesgue measure of the largest open and closed set contained in a subset of R

Question

Given an arbitrary set $E \subseteq R$

What is the relationship between the largest open and closed set contained in $E$?

If $O$ is the largest open set contained in $E$ and $C$ the largest closed set contained in $E$

IS $O \subset C$ or $C \subset O$?

More importantly, Is $\lambda(C) = \lambda(O)$?

where $\lambda$ is the Lebesgue measure!

Answer 1

Let $X\subseteq \Bbb R$ be a measurable set; then for each $\epsilon \geq\,0$ there exist an open set $A$ and a closed set $C$ such that $C\subseteq X \subseteq A $ and $$L^n ( A\setminus C )\leq \epsilon . $$

So that the outer set is the open one while inside $X $ there's the closed set.

You can find this property on wikipedia, under the 9th property of the Lebesgue measure: reference