Vitali's Theorem and non-existence of a measure

Vitali's construction of Non-measurable subset of $\mathbb {R}$

We define an equivalence relation on $[0,1]$ as follows:

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

Chose one representative of each class, to obtain a set $E$. Note that we need the axiom of choice to make this selection.

Any element $a$ in $[0,1]$ must be in a set $E+q$ ($E$ translated by $q$ where $-1\leq q\leq 1$)

This is because $E$ contains a representative from the equivalence class of $a$. Further if $q\neq s,E+q,E+s$ are disjoint.

Therefore a infinite but countable number of translations of E cover $[0,1]$ and the union of these translations lies in $[-1,2]$

But this implies $E$ is not measurable. If it had measure $m$, each of the translates would have measure $m$. If $m=0$, the measure of $[0,1]=0$. If $m>0$ the measure of $[-1,2]$ is infinite. Therefore E is not measurable.

This example is due to Vitali and E is called a Vitali set. As $E\in 2^{\mathbb {R} }$ we find that not all sets in $2^{\mathbb {R} }$ are measurable.

Interesting further exploration:

If we exclude the axiom of choice except on countable collections of non empty set we can develop alternate set theory models, where all sets are measurable. While other models using Zermelo Fraenkel Set theory seem more commonly referred, Bogachev gives this example using the axiom of determinacy:

Consider a game, $G_{A}$ of two players I and II, associated with every set A consisting of infinite sequences, $a=(a_{0},a_{1},,...)$ of natural numbers.\\ Player I writes a number $b_{0}\in \mathbb {N}$ , then player II writes a number $b_{1}\in \mathbb {N}$ and so on; the players know all the previous moves. If the obtained sequence $b=(b_{0},b_{1},...)$ belongs to A, then I wins, otherwise II wins. The set A and game $G_{A}$ are called determined if one of the players a winning strategy (i.e., a rule to make steps corresponding to the steps of the opposite side leading to victory). The axiom of determinacy (AD) is the statement that every set $A\in \mathbb {N^{\infty }}$ is determined. \\

Bogachev references Kanovei's nonstandard analysis wherein it is shown that this has measurability of all sets of reals as a consequence.

Reference

Vladimir I. Bogachev - Measure Theory Volume 1