Vitali set
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Vitali set is an elementary example of a set of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s that is not Lebesgue measurable
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

, found by . The Vitali theorem is the existence theorem
Existence theorem
In mathematics, an existence theorem is a theorem with a statement beginning 'there exist ..', or more generally 'for all x, y, ... there exist ...'. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not...

 that there are such sets. There are uncountably many Vitali sets, and their existence is proven on the assumption of the axiom of choice.

Measurable sets

Certain sets have a definite 'length' or 'mass'. For instance, the interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

 [0, 1] is deemed to have length 1; more generally, an interval [a, b], ab, is deemed to have length ba. If we think of such intervals as metal rods with uniform density, they likewise have well-defined masses. The set [0, 1] ∪ [2, 3] is composed of two intervals of length one, so we take its total length to be 2. In terms of mass, we have two rods of mass 1, so the total mass is 2.

There is a natural question here: if E is an arbitrary subset of the real line, does it have a 'mass' or 'total length'? As an example, we might ask what is the mass of the set of rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s, given that the mass of the interval [0, 1] is 1. The rationals are dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...

 in the reals, so any non negative value may appear reasonable.

However the closest generalization to mass is sigma additivity
Sigma additivity
In mathematics, additivity and sigma additivity of a function defined on subsets of a given set are abstractions of the intuitive properties of size of a set.- Additive set functions :...

, which gives rise to the Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...

. It assigns a measure of ba to the interval [a, b], but will assign a measure of 0 to the set of rational numbers because it is countable. Any set which has a well-defined Lebesgue measure is said to be "measurable", but the construction of the Lebesgue measure (for instance using Carathéodory's extension theorem
Carathéodory's extension theorem
In measure theory, Carathéodory's extension theorem states that any σ-finite measure defined on a given ring R of subsets of a given set Ω can be uniquely extended to the σ-algebra generated by R...

) does not make it obvious whether there exist non-measurable sets. The answer to that question involves the axiom of choice.

Construction and proof

A Vitali set is a subset of which, for each real number , contains exactly one number such that is rational. (This implies that is uncountable, and also that is irrational for any .) Such sets can be shown to exist given the axiom of choice.

To construct a Vitali set , consider the additive quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

 . Each element of this group is a "shifted copy" of the rational numbers: a set of the form for some ; that is, they are coset
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...

s of the rational numbers as a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

 of the real numbers under addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

. Thus, the elements of this group are subsets of R and partition R. There are uncountably many elements and each element is dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...

 in R. Since each element intersects [0,1], we can use the axiom of choice to choose a set containing exactly one representative out of each element of .

A Vitali set is non-measurable. To show this, we argue by contradiction and assume that is measurable. Let q1, q2, ... be an enumeration of the rational numbers in [−1, 1] (recall that the rational numbers are countable). From the construction of , note that the translated sets , k = 1, 2, ... are pairwise disjoint, and further note that . (To see the first inclusion, consider any real number r in [0,1] and let v be the representative in for the equivalence class [r]; then rv = q for some rational number q in [-1,1].)

Apply the Lebesgue measure to these inclusions using sigma additivity
Sigma additivity
In mathematics, additivity and sigma additivity of a function defined on subsets of a given set are abstractions of the intuitive properties of size of a set.- Additive set functions :...

:

Because the Lebesgue measure is translation invariant, and therefore

But this is impossible. Summing infinitely many copies of the constant yields either zero or infinity, according to whether the constant is zero or positive. In neither case is the sum in [1,3]. So cannot have been measurable after all, i.e., the Lebesgue measure λ must not define any value for .
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