View source for Non-measurable set

{{Short description|Set which cannot be assigned a meaningful "volume"}}
{{More citations needed|date=August 2009}}

In [[mathematics]], a '''non-measurable set''' is a [[Set (mathematics)|set]] which cannot be assigned a meaningful "volume".  The existence of such sets is construed to provide information about the notions of [[length]], [[area]] and [[volume]] in formal set theory. In [[Zermelo–Fraenkel set theory]], the [[axiom of choice]] entails that non-measurable subsets of <math>\mathbb{R}</math> exist.

The notion of a non-measurable set has been a source of great controversy since its introduction. Historically, this led [[Émile Borel|Borel]] and [[Kolmogorov]] to formulate probability theory on sets which are constrained to be measurable. The measurable sets on the line are iterated countable unions and intersections of intervals (called [[Borel set]]s) plus-minus [[null set]]s. These sets are rich enough to include every conceivable definition of a set that arises in standard mathematics, but they require a lot of formalism to prove that sets are measurable.

In 1970, [[Robert M. Solovay]] constructed the [[Solovay model]], which shows that it is consistent with standard set theory without uncountable choice,  that all subsets of the reals are measurable. However, Solovay's result depends on the existence of an [[inaccessible cardinal]], whose existence and consistency cannot be proved within standard set theory.

==Historical constructions==
The first indication that there might be a problem in defining length for an arbitrary set came from [[Vitali set|Vitali's theorem]].<ref>Moore, Gregory H., Zermelo's Axiom of Choice, Springer-Verlag, 1982, pp. 100–101</ref> A more recent combinatorial construction which is similar to the construction by Robin Thomas of a non-Lebesgue measurable set with some additional properties appeared in American Mathematical Monthly.<ref>{{Cite journal|last=Sadhukhan|first=A.|date=December 2022|title=A Combinatorial Proof of the Existence of Dense Subsets in <math>\mathbb{R}</math> without the "Steinhaus" like Property|journal=[[Am. Math. Mon.]]|language=en|volume=130|issue=2|pages=175|doi=10.1080/00029890.2022.2144665|arxiv=2201.03735}}</ref>

One would expect the measure of the union of two disjoint sets to be the sum of the measure of the two sets. A measure with this natural property is called ''finitely additive''. While a finitely additive measure is sufficient for most intuition of area, and is analogous to [[Riemann integration]], it is considered insufficient for [[probability]], because conventional modern treatments of sequences of events or random variables demand [[countable additivity]].

In this respect, the plane is similar to the line;  there is a finitely additive measure, extending Lebesgue measure, which is invariant under all [[isometries]]. For higher [[dimension]]s the picture gets worse. The [[Hausdorff paradox]] and [[Banach–Tarski paradox]] show that a three-dimensional [[ball (mathematics)|ball]] of radius 1 can be dissected into 5 parts which can be reassembled to form two balls of radius 1.

==Example==
Consider <math>S,</math> the set of all points in the unit circle, and the [[Group action (mathematics)|action]] on <math>S</math> by a group <math>G</math> consisting of all rational rotations (rotations by angles which are [[Rational number|rational]] multiples of <math>\pi</math>). Here <math>G</math> is countable (more specifically, <math>G</math> is isomorphic to <math>\Q/\Z</math>) while <math>S</math> is uncountable. Hence <math>S</math> breaks up into uncountably many [[Orbit (group theory)|orbits]] under <math>G</math> (the orbit of <math>s \in S</math> is the countable set <math>\{ s e^{i q \pi} : q \in \Q \}</math>). Using the [[axiom of choice]], we could pick a single point from each orbit, obtaining an uncountable subset <math>X \subset S</math> with the property that all of the rational translates (translated copies of the form <math>e^{i q \pi} X := \{ e^{i q \pi} x : x \in X \}</math> for some rational <math>q</math>)<ref>{{Cite journal|last=Ábrego|first=Bernardo M.|last2=Fernández-Merchant|first2=Silvia|last3=Llano|first3=Bernardo|date=January 2010|title=On the Maximum Number of Translates in a Point Set|journal=[[Discrete & Computational Geometry]]|language=en|volume=43|issue=1|pages=1–20|doi=10.1007/s00454-008-9111-9|issn=0179-5376|doi-access=free}}</ref> of <math>X</math> by <math>G</math> are [[pairwise disjoint]] (meaning, disjoint from <math>X</math> and from each other). The set of those translates [[Partition of a set|partitions]] the circle into a countable collection of disjoint sets, which are all pairwise congruent (by rational rotations). The set <math>X</math> will be non-measurable for any rotation-invariant countably additive probability measure on <math>S</math>: if <math>X</math> has zero measure, countable additivity would imply that the whole circle has zero measure.  If <math>X</math> has positive measure, countable additivity would show that the circle has infinite measure.

==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}}

==See also==

* {{annotated link|Banach–Tarski paradox}}
* {{annotated link|Carathéodory's criterion}}
* {{annotated link|Hausdorff paradox}}
* {{annotated link|Measure (mathematics)}}
* {{annotated link|Non-Borel set}}
* {{annotated link|Outer measure}}
* {{annotated link|Vitali set}}

==References==
===Notes===
{{reflist}}

===Bibliography===

{{refbegin}}
*{{cite journal|last=Dewdney|first=A. K.|date=1989|title=A matter fabricator provides matter for thought|journal=Scientific American|issue=April|pages=116&ndash;119|doi=10.1038/scientificamerican0489-116}}
{{refend}}

{{Measure theory}}

{{DEFAULTSORT:Non-Measurable Set}}
[[Category:Measure theory]]