We know that that a for measure $P$ on $(\mathbb{R},\mathcal{B}(\mathbb{R})$, for every measurable $A$ such that $P(A) < \infty$ there is a $G$ open and $F$ closed such that $F \subset A \subset G$ and $P(G-F)<\epsilon$. I m tring to see if the other way can be true ...
Is a $G$ open and $F$ closed such that $G \subset A \subset F$ and $P(F-G)<\epsilon$. ?
It is not true for the Lebesgue measure but can it be true for another measure.