Vitali set

Remarks

Let $q_1, q_2, \ldots$ list the countable set $\Q \cap [-1, 1]$, and put $V_k = V + q_k = \set{v + q_k : v \in V}$. The sets $V_k$ are pairwise disjoint: if $v + q_k = w + q_j$ with $v, w \in V$, then $v \sim w$, so $v = w$ and $q_k = q_j$. Every $x \in [0, 1]$ is equivalent to some $v \in V$, and then $x - v \in \Q \cap [-1, 1]$; so $[0, 1] \subseteq \bigcup_k V_k \subseteq [-1, 2]$. If $V$ were measurable, each $V_k$ would be measurable with $\lambda(V_k) = \lambda(V)$, by translation invariance. Countable additivity would give $1 \le \sum_{k} \lambda(V) \le 3$. This is impossible: the sum is $0$ if $\lambda(V) = 0$ and $\infty$ otherwise. The same argument shows that there is no translation-invariant, countably additive measure on all subsets of $\R$ that gives $[0, 1]$ a finite positive value. Some form of choice is needed. Solovay showed that, if an inaccessible cardinal is consistent with ZFC, then so is ZF with dependent choice and all sets of reals Lebesgue measurable.