Lebesgue Measure
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In
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
, a branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Lebesgue measure, named after French mathematician
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
, 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 * Mea ...
to
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of ''n''-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. For ''n'' = 1, 2, or 3, it coincides with the standard measure 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 Interna ...
,
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 A shape or figure is a graphics, graphical representation of an obje ...
, or
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). The de ...
. In general, it is also called ''n''-dimensional volume, ''n''-volume, or simply volume. It is used throughout
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include converg ...
, in particular to define
Lebesgue integration In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
. 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 of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
. Both were published as part of his dissertation in 1902.


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 field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer mea ...
\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 each element 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, 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 Zerm ...
s do exist; an example is the
Vitali set In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vita ...
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, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s is Lebesgue-measurable, and its Lebesgue measure is the length . The
open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
has the same measure, since the
difference Difference, The Difference, Differences or Differently may refer to: Music * ''Difference'' (album), by Dreamtale, 2005 * ''Differently'' (album), by Cassie Davis, 2009 ** "Differently" (song), by Cassie Davis, 2009 * ''The Difference'' (al ...
between the two sets consists only of the end points ''a'' and ''b'', which each have
measure zero 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 ...
. * Any
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
of intervals and is Lebesgue-measurable, and its Lebesgue measure is , the area of the corresponding
rectangle In 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 parallelogram containi ...
. * Moreover, every
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 named ...
is Lebesgue-measurable. However, there are Lebesgue-measurable sets which are not Borel sets. * Any
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
set of real numbers has Lebesgue measure 0. In particular, the Lebesgue measure of the set of
algebraic numbers An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
is 0, even though the set is
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
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 unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Thr ...
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 1 + \log_2(d) ~) no pair of integers ~(\,p,\,q\,)~ exists ...
s are examples of
uncountable set In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
s that have Lebesgue measure 0. * If the
axiom of determinacy In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of ...
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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
. *
Vitali set In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vita ...
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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
. *
Osgood curve In mathematical analysis, an Osgood curve is a non-self-intersecting curve that has positive area. Despite its area, it is not possible for such a curve to cover a convex set, distinguishing them from space-filling curves. Osgood curves are named ...
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 (ge ...
s with
positive Positive is a property of 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 "+" used to indicate a posi ...
Lebesgue measure (it can be obtained by small variation of the
Peano curve In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injecti ...
construction). The
dragon curve A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The dragon curve is probably most commonly thought of as the shape that is generated from repe ...
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, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
has a zero Lebesgue measure in its
ambient space An ambient space or ambient configuration space is the space surrounding an object. While the ambient space and hodological space are both considered ways of perceiving penetrable space, the former perceives space as ''navigable'', while the latt ...
.


Properties

The Lebesgue measure on R''n'' has the following properties: # If ''A'' is a
cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
of
intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval e ...
''I''1 × ''I''2 × ⋯ × ''I''''n'', then ''A'' is Lebesgue-measurable and \lambda (A)=, I_1, \cdot , I_2, \cdots , I_n, . # If ''A'' is a
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (th ...
of
countably many In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
disjoint Lebesgue-measurable sets, then ''A'' is itself Lebesgue-measurable and ''λ''(''A'') is equal to the sum (or
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
) of the measures of the involved measurable sets. # If ''A'' is Lebesgue-measurable, then so is its
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...
. # ''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.) # 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), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...
or closed subset of R''n'' (or even
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 named ...
, see
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
), 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. # 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 of a set can be approximated from within by compact subsets. Definition Let (''X'', ''T'') be a Hausdorff topological space and let Σ be a σ-algebra on ''X'' that ...
, and so it is a
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
. # Lebesgue measure is strictly positive on non-empty open sets, and so its support is the whole of R''n''. # If ''A'' is a Lebesgue-measurable set with ''λ(''A'') = 0 (a
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 ...
), then every subset of ''A'' is also a null set.
A fortiori ''Argumentum a fortiori'' (literally "argument from the stronger eason) (, ) is a form of argumentation that draws upon existing confidence in a proposition to argue in favor of a second proposition that is held to be implicit in, and even more cer ...
, 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, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
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) * 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, which implies t ...
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 * Mea ...
on that σ-algebra with \lambda( ,1times , 1times \cdots \times , 1=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, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
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 is zero, of ...
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, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occ ...
on R''n'' (or any metric Lipschitz equivalent to it). On the other hand, a set may have
topological dimension In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. Informal discussion For ordinary Euclidean ...
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 that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volt ...
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, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. Th ...
(''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, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets ''R'' of a given set ''Ω'' can be extended to a measure on the σ- ...
. 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 field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer mea ...
''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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
, which is independent from many of the conventional systems of axioms for
set theory 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, set theory, as a branch of mathematics, is mostly conce ...
. 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, 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 Zerm ...
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-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be p ...
. 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, 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 as ...
in the absence of the axiom of choice (see
Solovay's model In the mathematical field of set theory, the Solovay model is a model constructed by in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measur ...
).


Relation to other measures

The Borel measure 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) * 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, which implies t ...
. 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 was introduced by Alfréd Haar in 1933, though ...
can be defined on any
locally compact group In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are loc ...
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 as ...
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 ...
s, for example, surfaces or curves in R3 and
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
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 is zero, of ...
. 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 measurable set A\subset \R^n, the "density" of ''A'' is 0 or 1 at almost every point in \R^n. Additionally, the "density" of ''A'' is 1 at almost every point in ''A''. Intuit ...
* Lebesgue measure of the set of Liouville numbers *
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 Zerm ...
**
Vitali set In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vita ...


References

{{Measure theory Measures (measure theory)