Non-measurable set

In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical 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 $\mathbb {R}$ exist.

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. The measurable sets on the line are iterated countable unions and intersections of intervals (called Borel sets) plus-minus null sets. 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[edit]

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.^[2]

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 dimensions the picture gets worse. 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.

Example[edit]

Consider $S,$ the set of all points in the unit circle, and the action on $S$ by a group $G$ consisting of all rational rotations (rotations by angles which are rational multiples of $\pi$ ). Here $G$ is countable (more specifically, $G$ is isomorphic to $\mathbb {Q} /\mathbb {Z}$ ) while $S$ is uncountable. Hence $S$ breaks up into uncountably many orbits under $G$ (the orbit of $s\in S$ is the countable set $\{se^{iq\pi }:q\in \mathbb {Q} \}$ ). Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset $X\subset S$ with the property that all of the rational translates (translated copies of the form $e^{iq\pi }X:=\{e^{iq\pi }x:x\in X\}$ for some rational $q$ )^[3] of $X$ by $G$ are pairwise disjoint (meaning, disjoint from $X$ and from each other). The set of those translates partitions the circle into a countable collection of disjoint sets, which are all pairwise congruent (by rational rotations). The set $X$ will be non-measurable for any rotation-invariant countably additive probability measure on $S$ : if $X$ has zero measure, countable additivity would imply that the whole circle has zero measure. If $X$ has positive measure, countable additivity would show that the circle has infinite measure.