Editing Non-measurable set (section)

==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 five concessions is made:{{Citation needed|date=June 2023}}
# The volume of a set might change when it is rotated.
# The volume of the union of two disjoint sets might be different from the sum of their volumes.
# Some sets might be tagged "non-measurable", and one would need to check whether a set is "measurable" before talking about its volume.
# The axioms of ZFC ([[Zermelo–Fraenkel set theory]] with the axiom of choice) might have to be altered.
# The volume of <math>[0,1]^3</math> is <math>0</math> or <math>\infty</math>.

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.{{Citation needed|date=June 2023}} It is usually very easy to prove that a given specific subset of the geometric plane is measurable.{{Citation needed|date=June 2023}}  The fundamental assumption is that a countably infinite sequence of disjoint sets satisfies the sum formula, a property  called [[sigma additivity|σ-additivity]].

In 1970, [[Robert M. Solovay|Solovay]] demonstrated that the existence of a non-measurable set for the [[Lebesgue measure]] is not provable within the framework of Zermelo–Fraenkel set theory in the absence of an additional axiom (such as 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.{{Citation needed|date=June 2023}}

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]].{{Citation needed|date=June 2023}} It also affects the study of infinite groups to a large extent, as well as [[ring theory|ring]] and [[order theory]] (see [[Boolean prime ideal theorem]]).{{Citation needed|date=June 2023}} 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.{{Citation needed|date=June 2023}}