Non-measurable set
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This page gives a non-technical description of this concept. For a technical description see measure (mathematics) and the various constructions of non-measurable sets, Vitali set, Hausdorff paradox, Banach–Tarski paradox.

In mathematics, a non-measurable set is a subset of a set with finite positive measure where the subset's structure is so complicated that it cannot itself have a meaningful measure. Such sets are constructed to shed light on the notions of length, area and volume in formal set theory.

The notion of a non-measurable set has been a source of great controversy since its introduction. It is intuitively obvious to many people that any subset S of the unit disk (or unit line) has a measure, because one can throw darts at the disk (see Freiling's axiom of symmetry), and the probability of landing in S is the measure of the set. A non-measurable set means that it is inconsistent to talk about randomly picked geometric points.

Historically, this led Borel and Kolmogorov to formulate probability theory on sets which are constrained to be measurable. The measurable sets on the line are formed by countable unions and intersections of intervals. 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 1979, Solovay established that it is consistent with standard set theory, excluding uncountable choice, to assume that there are no non-measurable sets.

Historical constructions

The first indication that there might be a problem in defining length for an arbitrary set came from Vitali's theorem.^{citation needed}

A measure where adding up zero-measured sets you could get a set with measure 1 or 2 is called a finitely additive measure. 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. When you increase in dimension the picture gets worse. The Hausdorff paradox and Banach–Tarski paradox show that you can take a three dimensional ball of radius 1, dissect it into 5 parts, move and rotate the parts and get two balls of radius 1. Obviously this construction has no meaning in the physical world. In 1989, A. K. Dewdney published a letter from his friend Arlo Lipof in the Computer Recreations column of the Scientific American where he describes an underground operation "in a South American country" of doubling gold balls using the Banach-Tarski paradox.¹ Naturally, this was in the April issue, and "Arlo Lipof" is an anagram of "April Fool".

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 set theory (ZFC) might have to be altered

The mainstream approach is to take road 3. 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. This is the preferred option for most mathematicians. The fundamental assumption is that a countably infinite sequence of disjoint sets satisfies the sum formula, a property mathematicians call σ-additivity.

In terms of modifying ZFC to allow all sets to be measurable, the most common proposal by far is to remove the Axiom of Choice from ZFC (and hence work in ZF). In 1970, Solovay demonstrated that the existence of a non-measurable set for Lebesgue measure is not provable within the framework of Zermelo-Frankel set theory in the absence of the Axiom of Choice. He also showed that (assuming the consistency of an inaccessible cardinal) there is a model of ZF in which countable choice holds, every set is lebesgue measurable and in which the full axiom of choice fails.

Disallowing the axiom of choice seems to many mathematicians to eviscerate functional analysis and point-set topology, since 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 naively 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. But while option 4 is mathematicians' second most popular choice, it remains a very distant second.

Other mathematicians believe that one should give up σ-additivity in one dimension to get a definition of length for all sets. This has not proved to be very useful. A very short discussion of the reasons can be found in measure (mathematics).

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