Non-measurable Set - Consistent Definitions of Measure and Probability

Consistent Definitions of Measure and Probability

The Banach–Tarski paradox shows that there is no way to define volume in three dimensions unless one of the following four concessions is made:

1. The volume of a set might change when it is rotated
2. The volume of the union of two disjoint sets might be different from the sum of their volumes
3. Some sets might be tagged "non-measurable" and one would need to check if a set is "measurable" before talking about its volume
4. The axioms of ZFC (Zermelo–Fraenkel set theory with the axiom of Choice) might have to be altered

Standard measure theory takes the third option. One defines a family of measurable sets which is very rich, and almost any set explicitly defined in most branches of mathematics will be among this family. It is usually very easy to prove that a given specific subset of the geometric plane is measurable. The fundamental assumption is that a countably infinite sequence of disjoint sets satisfies the sum formula, a property called σ-additivity.

In 1970, Solovay demonstrated that the existence of a non-measurable set for Lebesgue measure is not provable within the framework of Zermelo–Fraenkel set theory in the absence of the Axiom of Choice, by showing that (assuming the consistency of an inaccessible cardinal) there is a model of ZF, called Solovay's model, in which countable choice holds, every set is Lebesgue measurable and in which the full axiom of choice fails.

The Axiom of Choice is equivalent to a fundamental result of point-set topology, Tychonoff's theorem, and also to the conjunction of two fundamental results of functional analysis, the Banach–Alaoglu theorem and the Krein–Milman theorem. It also affects the study of infinite groups to a large extent, as well as ring and order theory (see Boolean prime ideal theorem). However the axioms of determinacy and dependent choice, together, are sufficient for most geometric measure theory, potential theory, Fourier series and Fourier transforms, while making all subsets of the real line Lebesgue measurable.