Measurable subset of Vitaly set has measure zero. Proof.

$E_x = \{y \in [0,1]: x-y \in \Bbb{Q}\}$, $ \varepsilon=\{A \subset [0,1]: \exists x \quad A=E_x\} $ .We chose one element from each set of family $\varepsilon$. This is a Vitaly set $V$.

Prove that if $E$ is measurable and $E \subset V$ then $E$ has measure $0$.

$E_q = [0,1] \bigcap \Bbb{Q} $, $q \in \Bbb{Q} $

I don't know how $E$ looks. I know for example that every singleton is measurable and has measure zero. But I don't know how to explain that every measurable set of $V$ has measure zero.

Solution 1:

Consider

$$E_{\mathbb Q} = \bigcup_{\substack{r \in \mathbb Q \\ -1 \le r \le 1}} (E+r) \subseteq [-1,2]$$

This is a countable infinite union of disjoints subsets, each of those having the measure of $E$. If the measure of $E$ would be strictly positive, $E_{\mathbb Q}$ would have an infinite measure, in contradiction with $E_{\mathbb Q} \subseteq [-1,2]$.