In
mathematics, a non-measurable set is a
set which cannot be assigned a meaningful "volume". The
mathematical existence
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, whe ...
of such sets is construed to provide information about the notions of
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
,
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open su ...
and
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
in formal set theory. In
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
, the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
entails that non-measurable subsets of
exist.
The notion of a non-measurable set has been a source of great controversy since its introduction. Historically, this led
Borel and
Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
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
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are name ...
s) plus-minus
null set
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null s ...
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
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum o ...
, 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's theorem.
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
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of G ...
, it is considered insufficient for
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
, 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
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
s 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
Consider
the set of all points in the unit circle, and the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
on
by a group
consisting of all rational rotations (rotations by angles which are
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abil ...
multiples of
). Here
is countable (more specifically,
is isomorphic to
) while
is uncountable. Hence
breaks up into uncountably many
orbits
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...
under
(the orbit of
is the countable set
). Using the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, we could pick a single point from each orbit, obtaining an uncountable subset
with the property that all of the rational translates (translated copies of the form
for some rational
)
of
by
are
pairwise disjoint
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A ...
(meaning, disjoint from
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
will be non-measurable for any rotation-invariant countably additive probability measure on
: if
has zero measure, countable additivity would imply that the whole circle has zero measure. If
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:
# 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
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
with the axiom of choice) might have to be altered.
# The volume of
is
or
.
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. It is usually very easy to prove that a given specific subset of the geometric plane is measurable. The fundamental assumption is that a countably infinite sequence of disjoint sets satisfies the sum formula, a property called
σ-additivity
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity ...
.
In 1970,
Solovay
Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory.
Biography
Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation ...
demonstrated that the existence of a non-measurable set for the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
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
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum o ...
) there is a model of ZF, called
Solovay's model, in which
countable choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function ''A'' with domain N (where ...
holds, every set is Lebesgue measurable and in which the full axiom of choice fails.
The axiom of choice is equivalent to a fundamental result of
point-set topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
,
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is t ...
, and also to the conjunction of two fundamental results of functional analysis, the
Banach–Alaoglu theorem and the
Krein–Milman theorem. It also affects the study of infinite groups to a large extent, as well as
ring and
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
(see
Boolean prime ideal theorem
In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by consid ...
). However, the axioms of
determinacy
Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and simi ...
and
dependent choice together are sufficient for most
geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of ...
,
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
,
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
and
Fourier transforms
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
, while making all subsets of the real line Lebesgue-measurable.
See also
*
*
*
*
*
References
Notes
Bibliography
*
{{DEFAULTSORT:Non-Measurable Set
Measure theory