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Vitali set

A Vitali set is a subset of the real line, typically constructed within the unit interval [0,1], that serves as an explicit example of a set which is not Lebesgue measurable. Introduced by Italian mathematician Giuseppe Vitali in 1905, it demonstrates the existence of non-measurable sets under the standard Lebesgue measure on the reals, relying on the axiom of choice to ensure its construction.^[1] The construction begins by partitioning the real numbers into equivalence classes based on the relation x \sim y if and only if x - y is rational, forming the quotient group \mathbb{R}/\mathbb{Q}. Using the axiom of choice, one selects a single representative from each equivalence class intersected with [0,1] to form the Vitali set V, ensuring that V contains exactly one element from each such coset.^[1] This results in a set where the rational translates V + q for distinct rationals q are pairwise disjoint. To prove non-measurability, consider the countable collection of disjoint translates V + q for q \in \mathbb{Q} \cap [-1,1]; their union contains [0,1] and is contained in [-1,2].^[1] If V were Lebesgue measurable with measure m(V) > 0, the measure of the union would be infinite, exceeding the finite measure of [-1,2]; if m(V) = 0, the measure of [0,1] would be zero, contradicting the Lebesgue measure properties. Thus, no such measure exists for V, highlighting the limitations of Lebesgue measurability without additional axioms.^[1] The Vitali set's significance lies in its role as the first explicit construction of a non-measurable set, underscoring the axiom of choice's consequences in real analysis. In 1970, Robert Solovay showed that the existence of non-Lebesgue measurable sets is equivalent to the axiom of choice in certain models of set theory, confirming that Vitali sets cannot be constructed without it.^[2] Variations and generalizations of Vitali sets appear in studies of measure theory, Banach-Tarski paradox, and fair division problems in infinite settings.

Lebesgue Measure Basics

Definition of Measurable Sets

In the context of Lebesgue measure on the real line \mathbb{R}, the foundation begins with the Lebesgue outer measure m^*, which assigns a non-negative extended real number to every subset of \mathbb{R}. The outer measure of a set S \subseteq \mathbb{R} is defined as

m^*(S) = \inf \left\{ \sum_{I \in \mathcal{C}} \ell(I) \mid \mathcal{C} \text{ is a countable collection of open intervals covering } S \right\},

where \ell(I) denotes the length of the interval I. This definition ensures that m^* is translation-invariant, monotonic (if S \subseteq T, then m^*(S) \leq m^*(T)), and countably subadditive: m^*\left( \bigcup_{n=1}^\infty S_n \right) \leq \sum_{n=1}^\infty m^*(S_n) for any countable collection \{S_n\}. However, m^* is not countably additive on all subsets of \mathbb{R}, which motivates restricting to a subclass where additivity holds. A subset E \subseteq \mathbb{R} is Lebesgue measurable if it satisfies Carathéodory's criterion: for every test set T \subseteq \mathbb{R},

m^*(T \cap E) + m^*(T \cap E^c) = m^*(T),

where E^c = \mathbb{R} \setminus E is the complement of E.^[3] This condition ensures that E "splits" the outer measure of any set T additively, preventing overlaps or gaps in measurement. The collection of all Lebesgue measurable sets, denoted \mathcal{M}(\mathbb{R}), forms a \sigma-algebra: it includes \mathbb{R} and the empty set, is closed under complements, and is closed under countable unions (and thus countable intersections). On this \sigma-algebra, the Lebesgue measure m is defined by m(E) = m^*(E) for E \in \mathcal{M}(\mathbb{R}), and m is countably additive: for disjoint measurable sets \{E_k\}_{k=1}^\infty,

m\left( \bigcup_{k=1}^\infty E_k \right) = \sum_{k=1}^\infty m(E_k).[3]

All open intervals in \mathbb{R} are Lebesgue measurable, with m(I) = \ell(I) for any open interval I, and by extension, all Borel sets (generated by countable unions, intersections, and complements of open sets) are measurable. An equivalent characterization is that E is measurable if, for every \epsilon > 0, there exists an open set O \supseteq E such that m^*(O \setminus E) < \epsilon; the measure is then m(E) = \inf \{ m(O) \mid O \supseteq E, O \text{ open} \}.^[3] This approximation property highlights how measurable sets can be well-approximated by open sets from above and closed sets from below, distinguishing them from non-measurable sets like the Vitali set, which cannot satisfy these conditions without leading to contradictions in additivity.

Key Properties of Lebesgue Measure

The Lebesgue measure \mu on \mathbb{R}^d is a complete measure on the Lebesgue \sigma-algebra \mathcal{L}(\mathbb{R}^d), satisfying the standard axioms of a measure: it is non-negative, with \mu(E) \geq 0 for any measurable set E, and \mu(\emptyset) = 0.^[4] It exhibits countable additivity: for a countable collection of pairwise disjoint measurable sets \{E_k\}_{k=1}^\infty, \mu\left(\bigcup_{k=1}^\infty E_k\right) = \sum_{k=1}^\infty \mu(E_k).^[5] This property extends finite additivity and ensures the measure behaves consistently under countable unions of disjoint sets, a cornerstone for integration and analysis on \mathbb{R}^d.^[6] Monotonicity holds for measurable sets: if E \subseteq F and both are Lebesgue measurable with \mu(E) < \infty, then \mu(F \setminus E) = \mu(F) - \mu(E).^[4] The measure also satisfies continuity from below: for an increasing sequence of measurable sets E_1 \subseteq E_2 \subseteq \cdots, \mu\left(\bigcup_{k=1}^\infty E_k\right) = \lim_{k \to \infty} \mu(E_k).^[6] Continuity from above applies under finite measure: for a decreasing sequence E_1 \supseteq E_2 \supseteq \cdots with \mu(E_1) < \infty, \mu\left(\bigcap_{k=1}^\infty E_k\right) = \lim_{k \to \infty} \mu(E_k).^[4] These continuity properties facilitate limits in measure theory, such as in the monotone convergence theorem for integrals. A defining feature of Lebesgue measure is its translation invariance: for any measurable set E \subseteq \mathbb{R}^d and vector h \in \mathbb{R}^d, \mu(E + h) = \mu(E), where E + h = \{x + h : x \in E\}.^[5] This invariance extends from the outer measure, which is translation invariant for any set A \subseteq \mathbb{R}, ensuring \mu^*(A + y) = \mu^*(A) for the Lebesgue outer measure \mu^*.^[7] For intervals, \mu([a, b]) = b - a in one dimension, normalizing the measure on bounded intervals like [0,1] to have \mu([0,1]) = 1.^[4] Translation invariance is pivotal in constructions like the Vitali set, where rational translates cover intervals without overlap, leading to contradictions in measurability assumptions. Lebesgue measure is complete: every subset of a measure-zero set is measurable with measure zero.^[5] It is also regular, meaning for any measurable E and \varepsilon > 0, there exists an open set U \supseteq E with \mu(U \setminus E) < \varepsilon, and similarly for compact approximations from below when \mu(E) < \infty.^[4] These properties ensure the Lebesgue \sigma-algebra includes all Borel sets and is the completion thereof, providing a robust framework for \mathbb{R}^d while excluding non-measurable sets constructed via the axiom of choice.^[8]

Construction of the Vitali Set

Equivalence Relation on the Reals

The construction of the Vitali set begins with defining an equivalence relation on the set of real numbers \mathbb{R}, which partitions \mathbb{R} into disjoint equivalence classes. Specifically, two real numbers x and y are equivalent, denoted x \sim y, if and only if their difference x - y is a rational number, i.e., x - y \in \mathbb{Q}.^[9] This relation was introduced by Giuseppe Vitali in his 1905 paper "Sul problema della misura dei gruppi di punti di una retta" to facilitate the selection of representatives for constructing a non-measurable set.^[10]^[1] To verify that \sim is an equivalence relation, consider its properties. It is reflexive because for any x \in \mathbb{R}, x - x = 0 \in \mathbb{Q}. Symmetry holds since if x - y = q \in \mathbb{Q}, then y - x = -q \in \mathbb{Q}. Transitivity follows: if x \sim y and y \sim z, then x - z = (x - y) + (y - z) \in \mathbb{Q} + \mathbb{Q} = \mathbb{Q}. Thus, \sim partitions \mathbb{R} into equivalence classes, where each class is the set of all numbers differing from a fixed representative by a rational amount, forming cosets of the additive subgroup \mathbb{Q} in \mathbb{R}.^[11]^[9] Each equivalence class = \{x + q \mid q \in \mathbb{Q}\} is countable, as \mathbb{Q} is countable, and dense in \mathbb{R} because the rationals are dense. The partition consists of uncountably many such classes, since the quotient group \mathbb{R}/\mathbb{Q} has cardinality $2^{\aleph_0}, the continuum. In the Vitali construction, this relation enables the selection of a set containing exactly one element from each class within a bounded interval like [0, 1), using the axiom of choice.^[11]^[1]

Selection via the Axiom of Choice

The construction of a Vitali set proceeds by selecting exactly one representative from each equivalence class in the quotient group \mathbb{R}/\mathbb{Q}, where the equivalence classes are the sets of the form x + \mathbb{Q} for x \in \mathbb{R}. These classes partition the real line into uncountably many disjoint subsets, each dense in \mathbb{R} and countable. To form a Vitali set V, one typically restricts attention to the interval [0, 1) and chooses, for each equivalence class, a unique representative from its intersection with [0, 1), ensuring V \subseteq [0, 1). This selection process requires the axiom of choice (AC), as there is no explicit or constructive method to uniformly pick one element from each of the uncountably many nonempty sets (x + \mathbb{Q}) \cap [0, 1). AC asserts that for any collection of nonempty sets, there exists a choice function assigning to each set one of its elements. In this case, the family of sets is \{ (x + \mathbb{Q}) \cap [0, 1) \mid x + \mathbb{Q} \in \mathbb{R}/\mathbb{Q} \}, which is uncountable and lacks a natural ordering or definable selection rule due to the density of the rationals. Thus, AC guarantees the existence of V, though it provides no algorithm for its explicit construction.^[12] Giuseppe Vitali introduced this construction in 1905, implicitly relying on a principle akin to AC (formalized by Zermelo in 1904) to assert the existence of such a set without providing an explicit selection. Modern expositions emphasize that the Vitali set's non-constructive nature stems directly from AC, as alternative choice-free methods fail to produce it.

Proof of Non-Measurability

Covering by Rational Translates

The rational translates of the Vitali set V play a central role in demonstrating its non-measurability. Specifically, for any enumeration \{q_n\}_{n=1}^\infty of the rational numbers \mathbb{Q}, the sets V + q_n = \{v + q_n \mid v \in V\} are pairwise disjoint. To see this, suppose there exist v, v' \in V and distinct q_n, q_m \in \mathbb{Q} such that v + q_n = v' + q_m. Then v - v' = q_m - q_n \in \mathbb{Q}, which implies v \sim v' under the equivalence relation modulo \mathbb{Q}. However, since V contains exactly one representative from each equivalence class, it follows that v = v' and thus q_n = q_m, a contradiction. Moreover, these translates cover the entire real line: \mathbb{R} = \bigcup_{n=1}^\infty (V + q_n). For any x \in \mathbb{R}, let v \in V be the unique representative in the equivalence class of x, so x - v = q \in \mathbb{Q} for some q = q_n. Thus, x \in V + q_n. This disjoint countable covering exploits the translation invariance of Lebesgue measure, as each V + q_n has the same measure as V if V is measurable.^[1] To derive a contradiction within a bounded interval, restrict attention to the countable set C = \mathbb{Q} \cap [-1, 1] = \{r_k\}_{k=1}^\infty. The union U = \bigcup_{k=1}^\infty (V + r_k) satisfies [0, 1] \subseteq U \subseteq [-1, 2], assuming V \subseteq [0, 1]. The inclusion U \subseteq [-1, 2] holds because V + r_k \subseteq [r_k, 1 + r_k] \subseteq [-1, 2] for each r_k \in C. For the reverse, take any x \in [0, 1]; let v \in V be its representative, so q = x - v \in \mathbb{Q} and |q| \leq 1 (since both x, v \in [0, 1]), hence q = r_k \in C and x \in V + r_k \subseteq U. The sets V + r_k remain pairwise disjoint, as established previously.^[1]

Deriving the Contradiction

To derive the contradiction in the proof of non-measurability, assume for the sake of contradiction that the Vitali set V \subset [0,1] is Lebesgue measurable with Lebesgue measure m(V) = \mu.^[1] Let \{r_k\}_{k=1}^\infty be an enumeration of the countable set \mathbb{Q} \cap [-1,1]. The translated sets V + r_k = \{v + r_k : v \in V\} are pairwise disjoint, because if (V + r_i) \cap (V + r_j) \neq \emptyset for i \neq j, then there exist v_1, v_2 \in V such that v_1 + r_i = v_2 + r_j, implying v_1 - v_2 = r_j - r_i \in \mathbb{Q} and thus v_1 \sim v_2, contradicting the choice of distinct representatives in V.^[1] Moreover, the union \bigcup_{k=1}^\infty (V + r_k) contains [0,1], since every x \in [0,1] has representative v \in V with x = v + q for q \in \mathbb{Q} \cap [-1,1]. This union is contained in [-1,2].^[1] Since Lebesgue measure is translation-invariant, m(V + r_k) = \mu for each k, and by countable additivity (as the sets are disjoint and measurable under the assumption),

m\left( \bigcup_{k=1}^\infty (V + r_k) \right) = \sum_{k=1}^\infty \mu.

Thus,

$1 = m([0,1]) \leq \sum_{k=1}^\infty \mu \leq m([-1,2]) = 3.

If \mu > 0, the infinite sum diverges to \infty > 3, a contradiction. If \mu = 0, the sum is $0 < 1, again a contradiction. Therefore, the assumption that V is measurable must be false.^[1]

Further Properties

Cardinality and Topological Aspects

The cardinality of a Vitali set equals the cardinality of the continuum, $2^{\aleph_0}, as it consists of exactly one representative from each equivalence class in the quotient \mathbb{R}/\mathbb{Q}, and the number of such classes is likewise $2^{\aleph_0}.^[13] This follows from the fact that \mathbb{R} is the disjoint union of countably infinite equivalence classes, each of cardinality \aleph_0, so the index set for the classes must have cardinality $2^{\aleph_0} to yield the total cardinality of \mathbb{R}.^[13] Topologically, no Vitali set possesses the Baire property in \mathbb{R}.^[14] This property requires that a set either is meager (a countable union of nowhere dense sets) or has meager complement in every open interval, but the rational translates of a Vitali set V partition \mathbb{R} into countably many disjoint copies of V, and if V had the Baire property, this partition would imply a contradiction with the Baire category theorem, as \mathbb{R} is not meager.^[14] Vitali sets exhibit varied topological behaviors depending on the choice of representatives, enabled by the axiom of choice. For instance, there exist Vitali sets that are dense in \mathbb{R}, meaning their closure is all of \mathbb{R}, since each equivalence class is dense and selections can be made to ensure intersection with every nonempty open interval.^[15] Conversely, Vitali sets can also be constructed to be nowhere dense, though all such sets are unbounded and have empty interior, as no interval can be contained within a single equivalence class.^[15]

Measures in Other Contexts

While the Vitali set is non-measurable with respect to the Lebesgue measure, its measurability can differ under measures absolutely continuous with respect to Lebesgue measure on extended σ-algebras. Specifically, one can construct a measure μ that extends the Lebesgue measure λ on a larger σ-algebra containing the Vitali set V, such that μ(V) is defined and μ remains absolutely continuous with λ, rendering V measurable under μ.^[16] This extension is possible because absolute continuity does not preclude enlarging the domain of measurability beyond the Borel or Lebesgue σ-algebras.^[16] In such measure extensions, the assigned measure to a Vitali set can vary continuously from 0 to 1, depending on the construction. For instance, by rescaling the space or selecting Vitali sets within small intervals of Lebesgue measure ε, the measure μ(V) can be bounded above by ε and made arbitrarily close to any value in [0,1] while preserving absolute continuity.^[17] However, the possible measures form intervals like [0, x] or (0, x] for some x ≤ 1, reflecting constraints from the outer measure and the countable disjoint translates covering the unit interval.^[17] Regarding translation quasi-invariant measures on ℝ, the situation is mixed: certain Vitali subsets are measurable with respect to specific σ-finite quasi-invariant extensions of Lebesgue measure, while others remain nonmeasurable under every nonzero σ-finite translation quasi-invariant measure.^[18] Quasi-invariance, which requires that translates have measures differing by a positive density factor, allows for such variability without full translation invariance.^[18] In broader contexts, such as locally compact abelian groups equipped with Haar measure, Vitali-type sets—constructed via coset representatives modulo a countable dense subgroup—provide analogous examples of nonmeasurability. These sets exploit translation invariance to derive contradictions similar to the Lebesgue case, showing that no such selector is Haar measurable.^[19] This generalizes the original construction, highlighting the role of the axiom of choice in producing nonmeasurable selectors across group measure spaces.^[19]

Role of the Axiom of Choice

Dependence on AC

The construction of the Vitali set explicitly invokes the Axiom of Choice (AC) to define a set of representatives, one from each equivalence class in the partition of the real numbers \mathbb{R} under the relation x \sim y if and only if x - y \in \mathbb{Q}. This partition consists of uncountably many equivalence classes, each of cardinality |\mathbb{R}|, and selecting a transversal (a set intersecting each class exactly once) requires a choice function that AC provides for arbitrary families of non-empty sets. Without such a principle, no explicit method exists to construct the set in ZF set theory alone.^[20] Although the full AC suffices, the existence of the Vitali set—and hence non-Lebesgue measurable subsets of \mathbb{R}—is in fact a consequence of the strictly weaker Boolean Prime Ideal Theorem (BPI), which asserts that every Boolean algebra possesses a prime ideal. BPI implies the Hahn-Banach theorem over the reals and, more relevantly, the existence of ultrafilters that enable choice functions for the specific partition induced by \mathbb{Q}-cosets in \mathbb{R}. This connection was established through developments in choice principles following Zermelo's 1904 formulation of AC, with Vitali's 1905 construction implicitly relying on such a selection mechanism predating explicit AC notation.^[20] The necessity of AC or an equivalent principle is underscored by forcing models demonstrating its independence from ZF. In 1970, Robert M. Solovay constructed a model of ZF + the Axiom of Dependent Choices (DC) in which every subset of \mathbb{R} is Lebesgue measurable, assuming the consistency of an inaccessible cardinal; here, no Vitali set can exist, as its non-measurability would contradict the model's properties. This shows that the Vitali set's existence is not provable in ZF alone and requires augmenting the axioms with some form of choice. Solovay's result relies on collapsing the cardinals above an inaccessible one via forcing to ensure all sets of reals satisfy measurability, the Baire property, and the perfect set property.^[2] Conversely, certain models where full AC fails still admit Vitali sets. For instance, Paul Cohen's 1963 forcing construction yields a model of ZF where AC is false—specifically, there is no choice function for a countable family of pairs of reals—yet the continuum hypothesis holds, and BPI remains valid, allowing the construction of non-measurable sets like the Vitali set. These examples illustrate that while AC is neither necessary nor minimal for the Vitali set's existence, the absence of any choice principle beyond ZF consistently eliminates non-measurable sets of reals.

Alternative Models and Constructions

In models of set theory based on ZF without the full axiom of choice, the existence of a Vitali set can be avoided. A prominent example is Solovay's model, which assumes the existence of a strongly inaccessible cardinal and includes dependent choice (DC); in this model, every set of real numbers is Lebesgue measurable relative to the standard Borel structure on ℝ. Consequently, no Vitali set exists, as its construction and non-measurability rely on principles equivalent to some choice axiom. This demonstrates that the Vitali set's pathological properties are not theorems of ZF alone but depend on additional axioms for their realization. Alternative constructions of Vitali sets often generalize the original by varying the underlying equivalence relation or the ambient space. Instead of partitioning ℝ by cosets modulo ℚ, one can use other countable dense subgroups of (ℝ, +), such as the dyadic rationals ℤ[1/2] = {m/2^n : m ∈ ℤ, n ∈ ℕ}. Selecting a set of representatives for the cosets of ℤ[1/2] in ℝ via the axiom of choice yields a Vitali-type set V ⊆ [0,1), and the standard non-measurability proof applies: the countable disjoint union of its rational translates covers [0,1) up to measure zero but cannot have measure both zero and positive. Such variants produce distinct non-measurable sets, illustrating the multiplicity of pathological examples under ZFC.^[21] Further generalizations extend Vitali constructions to broader contexts, such as Polish groups equipped with σ-finite, translation-invariant Borel probability measures. In this framework, a Vitali-type set is a Borel selector for the cosets of a countable dense subgroup Γ, and under suitable regularity conditions on the measure, such a selector is non-measurable. Sławomir Solecki established key measurability properties for these sets, showing that no such selector can be universally measurable (measurable with respect to every σ-finite invariant measure) unless the group structure imposes additional constraints. These results unify the classical case with applications to non-abelian groups and abstract measure spaces, emphasizing the role of group actions in generating non-measurable sets.^[22]

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