Vitali set

The term Vitali set describes any set obtained by a particular mathematical construction. In fact, the construction uses the axiom of choice and the result given by an existence theorem is not uniquely determined. Vitali sets have many important applications in pure mathematics, most notable being a proof of existence of Lebesgue non-measurable sets in the measure theory. The name was given after the Italian mathematician Giuseppe Vitali.

Formal construction

We begin by defining the following relation on the real line. Two real numbers x and y are said to be equivalent if and only if the difference x-y is rational. In symbols,

$x\sim y\iff x-y\in \mathbb {Q} .$

It is easy to verify that it is in fact an equivalence relation. Thus, it yields the partition the set of reals into its equivalence classes. By the axiom of choice we can select a representative of each single class. The Vitali set V is defined to be the union all selected representatives.

Since for any real x the set $\{x+i:\;\;i\in \mathbb {Z} \}$ is in the same equivalence class, me may (and do) additionaly require that $V\subset [0,1].$