TITLE: Doubt about Exercise 5, Chapter 1, Shakarchi [RA]
QUESTION [5 upvotes]: So, I need to find an example of $E$ an open and bounded set, such that if $\mathcal{O}_n:=\{x \in E: d(x,E) < \frac
{1}{n}\}$
Then $\lim_{n \to \infty} m(\mathcal{O}_n) \neq m(E)$. Where $m$ is the Lebesgue measure.
My problem is the following, if all the $\mathcal{O}_n$ are measurable sets (Lebesgue measurable), you can apply Corolary 3.3 from the book, in other words, if $\mathcal{O}_n$ are all measurable, then the equality holds.
Because we are taking the limit of $m(\mathcal{O}_n)$, we need infinite many $\mathcal{O}_n$ to be not measurable, the only example of a non-measurable set is a Vitali set. So, how can a Vitali be the $\mathcal{O}_n$ of a set?
Any help would be appreciated.

REPLY [2 votes]: The idea is to make an assumption of Corollary 3.3 not satisfied. 
Let $\{x_n\}$ be an enumaration of $[0,1]\cap\mathbb{Q}$. Consider
$$
E=\cup_{n=1}^{\infty} (x_n-\frac 1{4^n}, x_n+\frac1{4^n}).$$
This is an open set with Lebesgue measure at most $\sum 2/4^n = 2/3$. 
Also, $E$ is dense in $[0,1]$. Then $m(O_n)\geq 1$ for each $n$.