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In
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
, a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the Lebesgue measure, named after
French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Republic (french: link=no, République française), is a transcontinental country This is a list of co ...

French
mathematician
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (a ...
, is the standard way of assigning a
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Measu ...
to
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
s of ''n''-dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
. For ''n'' = 1, 2, or 3, it coincides with the standard measure of
length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

length
,
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

area
, or
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

volume
. In general, it is also called ''n''-dimensional volume, ''n''-volume, or simply volume. It is used throughout
real analysis 200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

real analysis
, in particular to define
Lebesgue integration In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the -axis. The Lebesgue integral, ...
. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set ''A'' is here denoted by ''λ''(''A''). Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the
Lebesgue integral In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the -axis. The Lebesgue integral, ...
. Both were published as part of his dissertation in 1902. The Lebesgue measure is often denoted by ''dx'', but this should not be confused with the distinct notion of a
volume form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
.


Definition

For any interval I = ,b/math> (or I = (a, b)) in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq\mathbb, the Lebesgue
outer measure In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
\lambda^(E) is defined as an
infimum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to all elements of S, if such an element exists. Consequently, the term ''greatest low ...
:\lambda^(E) = \inf \left\. Some sets E satisfy the Carathéodory criterion, which requires that for every A\subseteq \mathbb, :\lambda^(A) = \lambda^(A \cap E) + \lambda^(A \cap E^c). The set of all such E forms a ''σ''-algebra. For any such E, its Lebesgue measure is defined to be its Lebesgue outer measure: \lambda(E) = \lambda^(E). A set E that does not satisfy the Carathéodory criterion is not Lebesgue-measurable.
Non-measurable set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s do exist; an example is the
Vitali set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s.


Intuition

The first part of the definition states that the subset E of the real numbers is reduced to its outer measure by coverage by sets of open intervals. Each of these sets of intervals I covers E in a sense, since the union of these intervals contains E. The total length of any covering interval set may overestimate the measure of E, because E is a subset of the union of the intervals, and so the intervals may include points which are not in E. The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E most tightly and do not overlap. That characterizes the Lebesgue outer measure. Whether this outer measure translates to the Lebesgue measure proper depends on an additional condition. This condition is tested by taking subsets A of the real numbers using E as an instrument to split A into two partitions: the part of A which intersects with E and the remaining part of A which is not in E: the set difference of A and E. These partitions of A are subject to the outer measure. If for all possible such subsets A of the real numbers, the partitions of A cut apart by E have outer measures whose sum is the outer measure of A, then the outer Lebesgue measure of E gives its Lebesgue measure. Intuitively, this condition means that the set E must not have some curious properties which causes a discrepancy in the measure of another set when E is used as a "mask" to "clip" that set, hinting at the existence of sets for which the Lebesgue outer measure does not give the Lebesgue measure. (Such sets are, in fact, not Lebesgue-measurable.)


Examples

* Any closed interval of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s is Lebesgue-measurable, and its Lebesgue measure is the length . The
open interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
has the same measure, since the
difference
difference
between the two sets consists only of the end points ''a'' and ''b'' and has
measure zero In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...
. * Any
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of intervals and is Lebesgue-measurable, and its Lebesgue measure is , the area of the corresponding
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

rectangle
. * Moreover, every
Borel set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borel sets. * Any
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
set of real numbers has Lebesgue measure 0. In particular, the Lebesgue measure of the set of
algebraic numbers An algebraic number is any complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, ev ...
is 0, even though the set is
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
in R. * The
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 188 ...

Cantor set
and the set of
Liouville number In number theory, a Liouville number is a real number ''x'' with the property that, for every positive integer ''n'', there exists a pair of integers (''p, q'') with ''q'' > 1 such that :0 \frac \ge \frac ~. Therefore, in the case ~ \left, c\,q ...
s are examples of
uncountable set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s that have Lebesgue measure 0. * If the
axiom of determinacy In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
holds then all sets of reals are Lebesgue-measurable. Determinacy is however not compatible with 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#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
. *
Vitali set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s are examples of sets that are not measurable with respect to the Lebesgue measure. Their existence relies on 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#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
. *
Osgood curve In mathematics, an Osgood curve is a non-self-intersecting curve (either a Jordan curve or a Jordan_curve_theorem#Definitions_and_the_statement_of_the_Jordan_theorem, Jordan arc) of positive area. More formally, these are curves in the Euclidean ...

Osgood curve
s are simple plane
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
s with
positive Positive is a property of Positivity (disambiguation), positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign " ...
Lebesgue measure (it can be obtained by small variation of the
Peano curve In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

Peano curve
construction). The
dragon curve A dragon curve is any member of a family of self-similar __NOTOC__ has an infinitely repeating self-similarity when it is magnified. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quanti ...

dragon curve
is another unusual example. * Any line in \mathbb^n, for n \geq 2, has a zero Lebesgue measure. In general, every proper
hyperplane In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
has a zero Lebesgue measure in its
ambient space An ambient space or ambient configuration space is the space Space is the boundless extent in which and events have relative and . In , physical space is often conceived in three s, although modern s usually consider it, with , to be part ...
.


Properties

The Lebesgue measure on R''n'' has the following properties: # If ''A'' is a
cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of intervals ''I''1 × ''I''2 × ⋯ × ''I''''n'', then ''A'' is Lebesgue-measurable and \lambda (A)=, I_1, \cdot , I_2, \cdots , I_n, . Here, , ''I'', denotes the length of the interval ''I''. # If ''A'' is a
disjoint union In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

disjoint union
of
countably many In mathematics, a Set (mathematics), set is countable if it has the same cardinality (the cardinal number, number of elements of the set) as some subset of the set of natural numbers N = . Equivalently, a set ''S'' is ''countable'' if there exist ...
disjoint Lebesgue-measurable sets, then ''A'' is itself Lebesgue-measurable and ''λ''(''A'') is equal to the sum (or
infinite series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
) of the measures of the involved measurable sets. # If ''A'' is Lebesgue-measurable, then so is its
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase add ...
. # ''λ''(''A'') ≥ 0 for every Lebesgue-measurable set ''A''. # If ''A'' and ''B'' are Lebesgue-measurable and ''A'' is a subset of ''B'', then ''λ''(''A'') ≤ ''λ''(''B''). (A consequence of 2, 3 and 4.) # Countable unions and intersections of Lebesgue-measurable sets are Lebesgue-measurable. (Not a consequence of 2 and 3, because a family of sets that is closed under complements and disjoint countable unions does not need to be closed under countable unions: \.) # If ''A'' is an
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australia ...
or
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
subset of R''n'' (or even
Borel set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, see
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
), then ''A'' is Lebesgue-measurable. # If ''A'' is a Lebesgue-measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure (see the
regularity theorem for Lebesgue measure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
). # A Lebesgue-measurable set can be "squeezed" between a containing open set and a contained closed set. This property has been used as an alternative definition of Lebesgue measurability. More precisely, E\subset \mathbb is Lebesgue-measurable if and only if for every \varepsilon>0 there exist an open set G and a closed set F such that F\subset E\subset G and \lambda(G\setminus F)<\varepsilon. # A Lebesgue-measurable set can be "squeezed" between a containing G''δ'' set and a contained F''σ''. I.e, if ''A'' is Lebesgue-measurable then there exist a G''δ'' set ''G'' and an F''σ'' ''F'' such that ''G'' ⊇ ''A'' ⊇ ''F'' and ''λ''(''G'' \ ''A'') = ''λ''(''A'' \ ''F'') = 0. # Lebesgue measure is both locally finite and
inner regular In mathematics, an inner regular measure is one for which the Measure (mathematics), measure of a set can be approximated from within by Compact space, compact subsets. Definition Let (''X'', ''T'') be a Hausdorff space, Hausdorff topological spa ...
, and so it is a
Radon measure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. # Lebesgue measure is strictly positive on non-empty open sets, and so its
support Support may refer to: Business and finance * Support (technical analysis) In stock market technical analysis, support and resistance are certain predetermined levels of the price of a security (finance), security at which it is thought that th ...
is the whole of R''n''. # If ''A'' is a Lebesgue-measurable set with ''λ(''A'') = 0 (a
null set In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...
), then every subset of ''A'' is also a null set.
A fortiori ''Argumentum a fortiori'' (literally "argument from the stronger") (, ) is a form of argumentation Argumentation theory, or argumentation, is the interdisciplinary study of how conclusions can be reached through logical reasoning; that is, claims ...
, every subset of ''A'' is measurable. # If ''A'' is Lebesgue-measurable and ''x'' is an element of R''n'', then the ''translation of ''A'' by x'', defined by ''A'' + ''x'' = , is also Lebesgue-measurable and has the same measure as ''A''. # If ''A'' is Lebesgue-measurable and \delta>0, then the ''dilation of A by \delta'' defined by \delta A=\ is also Lebesgue-measurable and has measure \delta^\lambda\,(A). # More generally, if ''T'' is a
linear transformation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and ''A'' is a measurable subset of R''n'', then ''T''(''A'') is also Lebesgue-measurable and has the measure \left, \det(T)\ \lambda(A). All the above may be succinctly summarized as follows (although the last two assertions are non-trivially linked to the following): : The Lebesgue-measurable sets form a ''σ''-algebra containing all products of intervals, and ''λ'' is the unique
complete Complete may refer to: Logic * Completeness (logic) * Complete theory, Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, ...
translation-invariant
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Measu ...
on that σ-algebra with \lambda( ,1times
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
times \cdots \times
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
=1. The Lebesgue measure also has the property of being ''σ''-finite.


Null sets

A subset of R''n'' is a ''null set'' if, for every ε > 0, it can be covered with countably many products of ''n'' intervals whose total volume is at most ε. All
countable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
sets are null sets. If a subset of R''n'' has
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point (geometry), po ...
less than ''n'' then it is a null set with respect to ''n''-dimensional Lebesgue measure. Here Hausdorff dimension is relative to the
Euclidean metric In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
on R''n'' (or any metric
Lipschitz Lipschitz, Lipshitz, or Lipchitz is an Ashkenazi Jewish surname. The surname has many variants, including: Lifshitz Lifshitz (or Lifschitz) is a surname, which may be derived from the Polish city of Głubczyce Głubczyce ( cs, Hlubčice or sparse ...
equivalent to it). On the other hand, a set may have
topological dimension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
less than ''n'' and have positive ''n''-dimensional Lebesgue measure. An example of this is the
Smith–Volterra–Cantor set In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set is an example of a set of points on the real line ℝ that is nowhere dense (in particular it contains no interval (mathematics), intervals), yet has positiv ...
which has topological dimension 0 yet has positive 1-dimensional Lebesgue measure. In order to show that a given set ''A'' is Lebesgue-measurable, one usually tries to find a "nicer" set ''B'' which differs from ''A'' only by a null set (in the sense that the
symmetric difference In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(''A'' − ''B'') ∪ (''B'' − ''A'') is a null set) and then show that ''B'' can be generated using countable unions and intersections from open or closed sets.


Construction of the Lebesgue measure

The modern construction of the Lebesgue measure is an application of
Carathéodory's extension theorem In measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in ...
. It proceeds as follows. Fix . A box in R''n'' is a set of the form :B=\prod_^n _i,b_i\, , where , and the product symbol here represents a Cartesian product. The volume of this box is defined to be :\operatorname(B)=\prod_^n (b_i-a_i) \, . For ''any'' subset ''A'' of R''n'', we can define its
outer measure In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
''λ''*(''A'') by: :\lambda^*(A) = \inf \left\ . We then define the set ''A'' to be Lebesgue-measurable if for every subset ''S'' of R''n'', :\lambda^*(S) = \lambda^*(S \cap A) + \lambda^*(S \setminus A) \, . These Lebesgue-measurable sets form a ''σ''-algebra, and the Lebesgue measure is defined by for any Lebesgue-measurable set ''A''. The existence of sets that are not Lebesgue-measurable is a consequence of the set-theoretical
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#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

axiom of choice
, which is independent from many of the conventional systems of axioms for
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
. The Vitali theorem, which follows from the axiom, states that there exist subsets of R that are not Lebesgue-measurable. Assuming the axiom of choice,
non-measurable set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s with many surprising properties have been demonstrated, such as those of the
Banach–Tarski paradox The Banach–Tarski paradox is a theorem in set theory, set-theoretic geometry, which states the following: Given a solid ball (mathematics), ball in 3‑dimensional space, existence theorem, there exists a decomposition of the ball into a finite ...
. In 1970, Robert M. Solovay showed that the existence of sets that are not Lebesgue-measurable is not provable within the framework of
Zermelo–Fraenkel set theory In set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set ...
in the absence of the axiom of choice (see Solovay's model).


Relation to other measures

The
Borel measure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
agrees with the Lebesgue measure on those sets for which it is defined; however, there are many more Lebesgue-measurable sets than there are Borel measurable sets. The Borel measure is translation-invariant, but not
complete Complete may refer to: Logic * Completeness (logic) * Complete theory, Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, ...
. The
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfré ...
can be defined on any
locally compact In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
group A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ident ...
and is a generalization of the Lebesgue measure (R''n'' with addition is a locally compact group). The
Hausdorff measure In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that ass ...
is a generalization of the Lebesgue measure that is useful for measuring the subsets of R''n'' of lower dimensions than ''n'', like
submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...
s, for example, surfaces or curves in R3 and
fractal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

fractal
sets. The Hausdorff measure is not to be confused with the notion of
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point (geometry), po ...
. It can be shown that there is no infinite-dimensional analogue of Lebesgue measure.


See also

*
Lebesgue's density theorem In mathematics, Lebesgue's density theorem states that for any Lebesgue measure, Lebesgue measurable set A\subset \R^n, the "density" of ''A'' is 0 or 1 at almost everywhere, almost every point in \R^n. Additionally, the "density" of ''A'' is 1 at a ...
* Liouville number#Liouville numbers and measure, Lebesgue measure of the set of Liouville numbers *
Non-measurable set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
**
Vitali set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...


References

{{Measure theory Measures (measure theory)