In Zermelo–Fraenkel set theory, the axiom of choice entails that non-measurable subsets of
The notion of a non-measurable set has been a source of great controversy since its introduction.
Historically, this led Borel and Kolmogorov to formulate probability theory on sets which are constrained to be 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.
The first indication that there might be a problem in defining length for an arbitrary set came from Vitali's theorem.
[1] 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.
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.
The Hausdorff paradox and Banach–Tarski paradox show that a three-dimensional ball of radius 1 can be dissected into 5 parts which can be reassembled to form two balls of radius 1.
the set of all points in the unit circle, and the action on
Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset
will be non-measurable for any rotation-invariant countably additive probability measure on
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] 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] It is usually very easy to prove that a given specific subset of the geometric plane is measurable.
[citation needed] 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 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] It also affects the study of infinite groups to a large extent, as well as ring and order theory (see Boolean prime ideal theorem).
[citation needed] 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.