Vitali's Theorem and non-existence of a measure

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Vitali's construction of Non-measurable subset of

We define an equivalence relation on as follows:

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

Any element in must be in a set ( translated by where )

This is because contains a representative from the equivalence class of . Further if are disjoint.

Therefore a infinite but countable number of translations of E cover and the union of these translations lies in

But this implies is not measurable. If it had measure , each of the translates would have measure . If , the measure of . If the measure of is infinite. Therefore E is not measurable.

This example is due to Vitali and E is called a Vitali set. As we find that not all sets in 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, of two players I and II, associated with every set A consisting of infinite sequences, of natural numbers.\\ Player I writes a number , then player II writes a number and so on; the players know all the previous moves. If the obtained sequence belongs to A, then I wins, otherwise II wins. The set A and game 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 is determined. \\

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


Vladimir I. Bogachev - Measure Theory Volume 1