Certain sets have a definite 'length' or 'mass'. For instance, the interval is deemed to have length 1; more generally, an interval , 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 ∪ 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, given that the mass of the interval is 1. The rationals are dense in the reals, so any non-negative value 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 , 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 does not make it obvious whether non-measurable sets exist. The answer to that question involves the axiom of choice.
A Vitali set is a subset of the interval of real numbers such that, for each real number, there is exactly one number such that is a rational number. Vitali sets exist because the rational numbers Q form a normal subgroup of the real numbers R under addition, and this allows the construction of the additive quotient groupR/Q of these two groups which is the group formed by the cosets of the rational numbers as a subgroup of the real numbers under addition. This group R/Q consists of disjoint "shifted copies" of Q in the sense that each element of this quotient group is a set of the form for some r in R. The uncountably many elements of R/Q partition R, and each element is dense in R. Each element of R/Q intersects , and the axiom of choice guarantees the existence of a subset of containing exactly one representative out of each element of R/Q. A set formed this way is called a Vitali set. Every Vitali set is uncountable, and is irrational for any.
Non-measurability
A Vitali set is non-measurable. To show this, we assume that V is measurable and we derive a contradiction. Let q1, q2,... be an enumeration of the rational numbers in . From the construction of V, 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 and let v be the representative in V for the equivalence class ; then r-v=qi for some rational number qi in which implies that r is in Vi. Apply the Lebesgue measure to these inclusions using sigma additivity: Because the Lebesgue measure is translation invariant, and therefore But this is impossible. Summing infinitely many copies of the constant λ yields either zero or infinity, according to whether the constant is zero or positive. In neither case is the sum in . So V cannot have been measurable after all, i.e., the Lebesgue measure λ must not define any value for λ.
