The Vitali set: a non-measurable subset of $\mathbb{R}$

Let $V \subseteq [0,1]$ contain exactly one element of each coset of $\mathbb{Q}$ in $\mathbb{R}$, that is, one representative of each class of the equivalence relation $x \sim y \iff x - y \in \mathbb{Q}$. Such a $V$ exists by the Axiom of Choice. Then $V$ is not Lebesgue measurable.

Indeed, enumerate the rationals of $[-1,1]$ as $q_1, q_2, \dots$ and put $V_k = V + q_k$. The $V_k$ are pairwise disjoint by the choice of one representative per class, and

$[0,1] \subseteq \bigcup_{k \ge 1} V_k \subseteq [-1, 2].$

If $V$ were measurable then so would each $V_k$ be, with $\lambda(V_k) = \lambda(V)$ by translation invariance, and countable additivity would give $1 \le \sum_{k} \lambda(V) \le 3$. The sum is $0$ if $\lambda(V) = 0$ and $+\infty$ otherwise, so neither case is possible.

The construction needs more than ZF + DC. By Solovay's theorem, if ZFC together with "there exists an inaccessible cardinal" is consistent, then there is a model of ZF + DC in which every subset of $\mathbb{R}$ is Lebesgue measurable. On that hypothesis ZF + DC does not prove that a Vitali set exists. The large-cardinal assumption is part of the claim and not a technicality: it is what Solovay's construction consumes, and by Shelah it cannot be removed.

Not proved in this library. It is recorded with citations and used in no proof here.

What would prove it. The countable additivity and translation invariance of Lebesgue measure (Lebesgue measure and the Lebesgue integral ‡), plus a choice function on the family of cosets (The Axiom of Choice ↗). The argument itself is three lines; the measure it argues about is what this library lacks.

Which page it serves. The order, Zorn and the Axiom of Choice page and its examples page, where the cost of choice is tracked result by result, and the Cantor set, Baire and measure zero page, which shows how badly a set can behave while remaining elementary. The Vitali set is the standard answer to "why is Lebesgue measure not defined on all of $\mathcal{P}(\mathbb{R})$", and it cannot be stated at all until a measure exists.

Where it sits in the choice ledger. It needs a choice principle strictly beyond dependent choice, which is a strong statement in a library that otherwise tracks choice carefully (The choice ledger: what costs the Axiom of Choice and what does not ↗). The relevant facts, all external here, are Solovay's model of ZF + DC in which every set of reals is Lebesgue measurable and has the Baire property, and Shelah's refinement, which shows that the inaccessible cardinal is genuinely needed for the measurability half though not for the Baire property half. Weaker principles than full choice already suffice: the Boolean prime ideal theorem yields a non-measurable set, by way of the Hahn-Banach theorem.