Vitali set

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 proof of their existence depends on the axiom of choice.

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.

There is a natural question here: if E is an arbitrary subset of the real line, does it have a 'mass' or 'total length'?

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.

Vitali sets exist because the rational numbers

form a normal subgroup of the real numbers

consists of disjoint "shifted copies" of

in the sense that each element of this quotient group is a set of the form

into disjoint sets, and each element is dense in

, and the axiom of choice guarantees the existence of a subset of

(recall that the rational numbers are countable).

Next, note that To see the first inclusion, consider any real number

Apply the Lebesgue measure to these inclusions using sigma additivity: Because the Lebesgue measure is translation invariant,

Summing infinitely many copies of the constant

yields either zero or infinity, according to whether the constant is zero or positive.

No Vitali set has the property of Baire.

[2] By modifying the above proof, one shows that each Vitali set has Banach measure 0.

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.

[3] 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.

[4] In 1980, Saharon Shelah proved that it is not possible to establish Solovay's result without his assumption on inaccessible cardinals.