In measure theory, the **Vitali set** is a typical, elementary example of a non-measurable set.

Despite the terminology, there are many Vitali sets. They are constructed using the axiom of choice, and for reasons too complex to discuss here, Vitali sets are impossible to describe explicitly.

*See also: Banach-Tarski paradox*

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, they likewise have well-defined masses. If the [0,1] rod weighs 1 kilogram, then the [3,9] rod weighs 6 kilograms. 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'd 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 "length?" As an example, we might ask what is the mass of the set of rational numbers. They are very finely spread over all of the real line, so any answer may appear reasonable at first pass.

As it turns out, the physically relevant solution is to use measure theory. In this setting, the Lebesgue measure, which assigns weight b-a to the interval [a,b] will assign weight 0 to the set of rational numbers. Any set which has a well-defined weight is said to be "measurable." The construction of the Lebesgue measure (for instance, using the outer measure) does not make obvious whether there are non-measurable sets.

Our construction and proof will proceed by contradiction. As a first step, we will construct a set V and assume that it is measurable. Our second step is to show that either translation invariance or countable additivity is violated. Since translation invariance and countable additivity are definetly true for the Lebesgue measure μ, it must be that our initial assumption is incorrect; namely, the set V must not be measurable.

The construction has many parallels to the construction of the paradoxical decompositions in the Banach-Tarski paradox.

If x,y are real numbers and x-y is a rational number, then we write x~y and we say that x and y are *equivalent*; ~ is an equivalence relation. For each x, there is a subset [x]={y∈**R**;x~y} called the *equivalence class* of x. The set of equivalent classes partitions **R**. By the axiom of choice, we are able to choose a set V⊂[0,1] such that for any equivalence class [x], the set V∩[x] is a singleton, that is, a set consisting of exactly one point.

V is the Vitali set. Note that there are in fact several choices of V; the axiom of choice lets you say there is such a V, but there are clearly infinitely many.

First we let x_{1},x_{2},... be an enumeration of the rational numbers in [-1,1] (Recall that the rational numbers are countable.) From the construction of V, note that the sets V_{k}=V+x_{k}, k=1,2,... are pairwise disjoint, and further note that [0,1]⊂∪V_{k}⊂[-1,2]. Because μ is countably additive, it must also have the propriety of being *monotone*; that is, if A⊂B, then μ(A)≤μ(B). Hence, we know that

- 1≤μ(∪V
_{k})≤3 (*)

- 1≤∑
_{k=1}^{∞}μ(V)≤3

This conclusion is absurd, and since all we've used is translation invariance and countable additivity, it must be that V is non-measurable.