Set of real numbers that is not Lebesgue measurable
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. Each Vitali set is uncountable, and there are uncountably many Vitali sets. The proof of their existence depends on the axiom of choice. 
Measurable sets.
Certain sets have a definite 'length' or 'mass'. For instance, the interval [0, 1] is deemed to have length 1; more generally, an interval ["a", "b"], "a" ≤ "b", is deemed to have length "b" − "a". 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 numbers between 0 and 1, given that the mass of the interval [0, 1] is 1. The rationals are dense in the reals, so any value between and including 0 and 1 may appear reasonable.
However the closest generalization to mass is sigma additivity, which gives rise to the Lebesgue measure. It assigns a measure of "b" − "a" 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) does not make it obvious whether non-measurable sets exist. The answer to that question involves the axiom of choice.
Construction and proof.
A Vitali set is a subset formula_1 of the interval formula_2 of real numbers such that, for each real number formula_3, there is exactly one number formula_4 such that formula_5 is a rational number. Vitali sets exist because the rational numbers formula_6 form a normal subgroup of the real numbers formula_7 under addition, and this allows the construction of the additive quotient group formula_8 of these two groups which is the group formed by the cosets formula_9 of the rational numbers as a subgroup of the real numbers under addition. This group formula_8 consists of disjoint "shifted copies" of formula_6 in the sense that each element of this quotient group is a set of the form formula_9 for some formula_3 in formula_7. The uncountably many elements of formula_8 partition formula_7 into disjoint sets, and each element is dense in formula_7. Each element of formula_8 intersects formula_2, and the axiom of choice guarantees the existence of a subset of formula_2 containing exactly one representative out of each element of formula_8. A set formed this way is called a Vitali set.
Every Vitali set formula_1 is uncountable, and formula_23 is irrational for any formula_24.
Non-measurability.
A Vitali set is non-measurable. To show this, we assume that formula_1 is measurable and we derive a contradiction. Let formula_26 be an enumeration of the rational numbers in formula_27 (recall that the rational numbers are countable). From the construction of formula_1, we can show that the translated sets formula_29, formula_30 are pairwise disjoint. (If not, then there exists distinct formula_31 and formula_32 such that formula_33, a contradiction.) 
Next, note that 
formula_34
To see the first inclusion, consider any real number formula_3 in formula_2 and let formula_37 be the representative in formula_1 for the equivalence class formula_39; then 
formula_40 for some rational number formula_41 in formula_27 which implies that formula_3 is in formula_44.
Apply the Lebesgue measure to these inclusions using sigma additivity:
formula_45
Because the Lebesgue measure is translation invariant, formula_46 and therefore
formula_47
But this is impossible. Summing infinitely many copies of the constant formula_48 yields either zero or infinity, according to whether the constant is zero or positive. In neither case is the sum in formula_49. So formula_1 cannot have been measurable after all, i.e., the Lebesgue measure formula_51 must not define any value for formula_48.
Properties.
No Vitali set has the property of Baire. 
By modifying the above proof, one shows that each Vitali set has Banach measure 0. This does not create any contradictions since Banach measures are not countably additive, but only finitely additive.
Role of the axiom of choice.
The construction of Vitali sets given above uses the axiom of choice. The question arises: is the axiom of choice needed to prove the existence of sets that are not Lebesgue measurable? The answer is yes, provided that inaccessible cardinals are consistent with the most common axiomatization of set theory, so-called ZFC.
In 1964, Robert Solovay constructed a model of Zermelo–Fraenkel set theory without the axiom of choice where all sets of real numbers are Lebesgue measurable. This is known as the Solovay model. In his proof, Solovay assumed that the existence of inaccessible cardinals is consistent with the other axioms of Zermelo-Fraenkel set theory, i.e. that it creates no contradictions. This assumption is widely believed to be true by set theorists, but it cannot be proven in ZFC alone.
In 1980, Saharon Shelah proved that it is not possible to establish Solovay's result without his assumption on inaccessible cardinals.
References.
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