Measure (mathematics)

Encyclopedia

In mathematical analysis

, a

a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure

on a Euclidean space

, which assigns the conventional length

, area

, and volume

of Euclidean geometry

to suitable subsets of the n-dimensional Euclidean space

[0, 1] in the real numbers

is its length in the everyday sense of the word, specifically 1.

To qualify as a measure (see Definition below), a function that assigns a non-negative real number or +∞

to a set's subsets must satisfy a few conditions. One important condition is countable additivity. This condition states that the size of the union of a sequence of disjoint subsets is equal to the sum of the sizes of the subsets. However, it is in general impossible to associate a consistent size to each subset of a given set and also satisfy the other axioms of a measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the subsets on which the measure is to be defined are called measurable and they are required to form a σ-algebra

, meaning that unions

, intersections

and complements

of sequences of measurable subsets are measurable. Non-measurable set

s in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated, in the sense of being badly mixed up with their complements; indeed, their existence is a non-trivial consequence of the axiom of choice.

Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel

, Henri Lebesgue

, Johann Radon

and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov

's axiomatisation of probability theory

and in ergodic theory

. In integration theory, specifying a measure allows one to define integral

s on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral

. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory

considers measures that are invariant under, or arise naturally from, a dynamical system

.

over a set . A function

from to the extended real number line

is called a

One may require that at least one set E has finite measure. Then the null set automatically has measure zero because of countable additivity, because and is finite if and only if the empty set has measure zero.

The pair is called a

is called a

If only the second and third conditions of the definition of measure above are met, and takes on at most one of the values , then is called a

A probability measure

is a measure with total measure one (i.e.); a probability space

is a measure space with a probability measure.

For measure spaces that are also topological space

s various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory

) are Radon measure

s. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous function

s with compact support. This approach is taken by Bourbaki

(2004) and a number of other authors. For more details see Radon measure

.

: If E

A measure μ is continuous from below: If E

of the sets E

of the sets E

This property is false without the assumption that at least one of the E

which all have infinite Lebesgue measure, but the intersection is empty.

A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). It is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. A set in a measure space has σ-finite measure if it is a countable union of sets with finite measure.

For example, the real number

s with the standard Lebesgue measure

are σ-finite but not finite. Consider the closed intervals [k,k+1] for all integer

s k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real number

s with the counting measure

, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property

of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.

A measurable set X is called a null set

if μ(X)=0. A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable.

A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference

of X and Y is contained in a null set. One defines μ(Y) to equal μ(X).

For any set I and any set of nonnegative r

A measure on is -additive if for any and any family , the following hold:

Note that the second condition is equivalent to the statement that the ideal

of null sets is -complete.

Other 'named' measures used in various theories include: Borel measure, Jordan measure

, ergodic measure, Euler measure, Gaussian measure

, Baire measure

, Radon measure

and Young measure

.

In physics an example of a measure is spatial distribution of mass

(see e.g., gravity potential), or another non-negative extensive property, conserved

(see conservation law

for a list of these) or not. Negative values lead to signed measures, see "generalizations" below.

Liouville measure, known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics.

Gibbs measure

is widely used in statistical mechanics, often under the name canonical ensemble

.

If the axiom of choice is assumed to be true, not all subsets of Euclidean space

are Lebesgue measurable; examples of such sets include the Vitali set

, and the non-measurable sets postulated by the Hausdorff paradox

and the Banach–Tarski paradox

.

with values in the (signed) real numbers is called a signed measure

, while such a function with values in the complex number

s is called a complex measure

. Measures that take values in Banach spaces have been studied extensively. A measure that takes values in the set of self-adjoint projections on a Hilbert space

is called a projection-valued measure

; these are used in functional analysis

for the spectral theorem

. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term

but not general linear combination

, while signed measures are the linear closure of positive measures.

Another generalization is the finitely additive measure, which are sometimes called contents. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first, but proved to be not so useful. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of L

and the Stone–Čech compactification

. All these are linked in one way or another to the axiom of choice.

A charge

is a generalization in both directions: it is a finitely additive, signed measure.

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, a

**measure**on a set is a systematic way to assign to each suitable subsetSubset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure

Lebesgue measure

In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...

on a Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, which assigns the conventional length

Length

In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

, area

Area

Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

, and volume

Volume

Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

of Euclidean geometry

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

to suitable subsets of the n-dimensional Euclidean space

**R**^{n}. For instance, the Lebesgue measure of the intervalInterval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

[0, 1] in the real numbers

Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

is its length in the everyday sense of the word, specifically 1.

To qualify as a measure (see Definition below), a function that assigns a non-negative real number or +∞

Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...

to a set's subsets must satisfy a few conditions. One important condition is countable additivity. This condition states that the size of the union of a sequence of disjoint subsets is equal to the sum of the sizes of the subsets. However, it is in general impossible to associate a consistent size to each subset of a given set and also satisfy the other axioms of a measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the subsets on which the measure is to be defined are called measurable and they are required to form a σ-algebra

Sigma-algebra

In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...

, meaning that unions

Union (set theory)

In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...

, intersections

Intersection (set theory)

In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

and complements

Complement (set theory)

In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...

of sequences of measurable subsets are measurable. Non-measurable set

Non-measurable set

In mathematics, a non-measurable set is a set whose structure is so complicated that it cannot be assigned any meaningful measure. Such sets are constructed to shed light on the notions of length, area and volume in formal set theory....

s in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated, in the sense of being badly mixed up with their complements; indeed, their existence is a non-trivial consequence of the axiom of choice.

Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel

Émile Borel

Félix Édouard Justin Émile Borel was a French mathematician and politician.Borel was born in Saint-Affrique, Aveyron. Along with René-Louis Baire and Henri Lebesgue, he was among the pioneers of measure theory and its application to probability theory. The concept of a Borel set is named in his...

, Henri Lebesgue

Henri Lebesgue

Henri Léon Lebesgue was a French mathematician most famous for his theory of integration, which was a generalization of the seventeenth century concept of integration—summing the area between an axis and the curve of a function defined for that axis...

, Johann Radon

Johann Radon

Johann Karl August Radon was an Austrian mathematician. His doctoral dissertation was on calculus of variations .- Life :...

and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov

Andrey Kolmogorov

Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...

's axiomatisation of probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

and in ergodic theory

Ergodic theory

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....

. In integration theory, specifying a measure allows one to define integral

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

s on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral

Riemann integral

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. The Riemann integral is unsuitable for many theoretical purposes...

. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory

Ergodic theory

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....

considers measures that are invariant under, or arise naturally from, a dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

.

## Definition

Let be a σ-algebraSigma-algebra

In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...

over a set . A function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

from to the extended real number line

Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...

is called a

**measure**if it satisfies the following properties:**Non-negativity**: for all

**Countable additivity**(or σ-additivitySigma additivityIn mathematics, additivity and sigma additivity of a function defined on subsets of a given set are abstractions of the intuitive properties of size of a set.- Additive set functions :...

): For all countable collections of pairwise disjoint sets in :

**Null empty set**:

One may require that at least one set E has finite measure. Then the null set automatically has measure zero because of countable additivity, because and is finite if and only if the empty set has measure zero.

The pair is called a

**measurable space**, the members of are called**measurable sets**, and the tripleTuple

In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...

is called a

**measure space**.If only the second and third conditions of the definition of measure above are met, and takes on at most one of the values , then is called a

**signed measure**

.Signed measure

In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Some authors may call it a charge, by analogy with electric charge, which is a familiar distribution that takes on positive and negative values.-Definition:There are two slightly...

A probability measure

Probability measure

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...

is a measure with total measure one (i.e.); a probability space

Probability space

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

is a measure space with a probability measure.

For measure spaces that are also topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

) are Radon measure

Radon measure

In mathematics , 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 locally finite and inner regular.-Motivation:...

s. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous function

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

s with compact support. This approach is taken by Bourbaki

Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...

(2004) and a number of other authors. For more details see Radon measure

Radon measure

In mathematics , 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 locally finite and inner regular.-Motivation:...

.

## Properties

Several further properties can be derived from the definition of a countably additive measure.### Monotonicity

A measure μ is monotonicMonotonic function

In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

: If E

_{1}and E_{2}are measurable sets with E_{1}⊆ E_{2}then### Measures of infinite unions of measurable sets

A measure μ is countably subadditive: If E_{1}, E_{2}, E_{3}, … is a countable sequence of sets in Σ, not necessarily disjoint, thenA measure μ is continuous from below: If E

_{1}, E_{2}, E_{3}, … are measurable sets and E_{n}is a subset of E_{n + 1}for all n, then the unionUnion (set theory)

In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...

of the sets E

_{n}is measurable, and### Measures of infinite intersections of measurable sets

A measure μ is continuous from above: If E_{1}, E_{2}, E_{3}, … are measurable sets and E_{n + 1}is a subset of E_{n}for all n, then the intersectionIntersection (set theory)

In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

of the sets E

_{n}is measurable; furthermore, if at least one of the E_{n}has finite measure, thenThis property is false without the assumption that at least one of the E

_{n}has finite measure. For instance, for each n ∈**N**, letwhich all have infinite Lebesgue measure, but the intersection is empty.

## Sigma-finite measures

A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). It is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. A set in a measure space has σ-finite measure if it is a countable union of sets with finite measure.

For example, the real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s with the standard Lebesgue measure

Lebesgue measure

In measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...

are σ-finite but not finite. Consider the closed intervals [k,k+1] for all integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s with the counting measure

Counting measure

In mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be the number of elements in the subset, if the subset is finite, and ∞ if the subset is infinite....

, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property

Lindelöf space

In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....

of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.

## Completeness

A measurable set X is called a null set

Null set

In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...

if μ(X)=0. A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable.

A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference

Symmetric difference

In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....

of X and Y is contained in a null set. One defines μ(Y) to equal μ(X).

## Additivity

Measures are required to be countably additive. However, the condition can be strengthened as follows.For any set I and any set of nonnegative r

_{i}, define:A measure on is -additive if for any and any family , the following hold:

Note that the second condition is equivalent to the statement that the ideal

Ideal (set theory)

In the mathematical field of set theory, an ideal is a collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal , and the union of any two elements of the ideal must also be in the ideal.More formally, given a set X, an...

of null sets is -complete.

## Examples

Some important measures are listed here.- The counting measureCounting measureIn mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be the number of elements in the subset, if the subset is finite, and ∞ if the subset is infinite....

is defined by μ(S) = number of elements in S. - The Lebesgue measureLebesgue measureIn measure theory, 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 with the standard measure of length, area, or volume. In general, it is also called...

on**R**is a complete translation-invariant measure on a σ-algebra containing the intervalInterval (mathematics)In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

s in**R**such that μ([0,1]) = 1; and every other measure with these properties extends Lebesgue measure. - Circular angleAngleIn geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

measure is invariant under rotationRotationA rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

, and hyperbolic angleHyperbolic angleIn mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle...

measure is invariant under squeeze mappingSqueeze mappingIn linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a Euclidean motion.For a fixed positive real number r, the mapping →...

. - The Haar measureHaar measureIn mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....

for a locally compactLocally compact spaceIn topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.-Formal definition:...

topological groupTopological groupIn mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...

is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties. - The Hausdorff measureHausdorff measureIn mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in Rn or, more generally, in any metric space. The zero dimensional Hausdorff measure is the number of points in the set or ∞ if the set is infinite...

is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets. - Every probability spaceProbability spaceIn probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit intervalUnit intervalIn mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...

[0,1]). Such a measure is called a probability measure. See probability axiomsProbability axiomsIn probability theory, the probability P of some event E, denoted P, is usually defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov, which are described below....

. - The Dirac measure δ
_{a}(cf. Dirac delta functionDirac delta functionThe Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

) is given by δ_{a}(S) = χ_{S}(a), where χ_{S}is the characteristic function of S. The measure of a set is 1 if it contains the point a and 0 otherwise.

Other 'named' measures used in various theories include: Borel measure, Jordan measure

Jordan measure

In mathematics, the Peano–Jordan measure is an extension of the notion of size to shapes more complicated than, for example, a triangle, disk, or parallelepiped....

, ergodic measure, Euler measure, Gaussian measure

Gaussian measure

In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces...

, Baire measure

Baire measure

A Baire measure is a measure on the σ-algebra of Baire sets of a topological space. In spaces that are not metric spaces, the Borel sets and the Baire sets may differ...

, Radon measure

Radon measure

In mathematics , 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 locally finite and inner regular.-Motivation:...

and Young measure

Young measure

In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. Young measures have applications in the calculus of variations and the study of nonlinear partial differential equations...

.

In physics an example of a measure is spatial distribution of mass

Mass

Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

(see e.g., gravity potential), or another non-negative extensive property, conserved

Conserved quantity

In mathematics, a conserved quantity of a dynamical system is a function H of the dependent variables that is a constant along each trajectory of the system. A conserved quantity can be a useful tool for qualitative analysis...

(see conservation law

Conservation law

In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....

for a list of these) or not. Negative values lead to signed measures, see "generalizations" below.

Liouville measure, known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics.

Gibbs measure

Gibbs measure

In mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is the measure associated with the Boltzmann distribution, and generalizes the notion of the canonical ensemble...

is widely used in statistical mechanics, often under the name canonical ensemble

Canonical ensemble

The canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system...

.

## Non-measurable sets

If the axiom of choice is assumed to be true, not all subsets of Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

are Lebesgue measurable; examples of such sets include the Vitali set

Vitali set

In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by . The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence is proven on the assumption of the axiom of...

, and the non-measurable sets postulated by the Hausdorff paradox

Hausdorff paradox

In mathematics, the Hausdorff paradox, named after Felix Hausdorff, states that if you remove a certain countable subset of the sphere S2, the remainder can be divided into three disjoint subsets A, B and C such that A, B, C and B ∪ C are all congruent...

and the Banach–Tarski paradox

Banach–Tarski paradox

The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...

.

## Generalizations

For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set functionSet function

In mathematics, a set function is a function whose input is a set. The output is usually a number. Often the input is a set of real numbers, a set of points in Euclidean space, or a set of points in some measure space.- Examples :...

with values in the (signed) real numbers is called a signed measure

Signed measure

In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Some authors may call it a charge, by analogy with electric charge, which is a familiar distribution that takes on positive and negative values.-Definition:There are two slightly...

, while such a function with values in the complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s is called a complex measure

Complex measure

In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size is a complex number.-Definition:...

. Measures that take values in Banach spaces have been studied extensively. A measure that takes values in the set of self-adjoint projections on a Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

is called a projection-valued measure

Projection-valued measure

In mathematics, particularly functional analysis a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a Hilbert space...

; these are used in functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

for the spectral theorem

Spectral theorem

In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...

. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term

**positive measure**is used. Positive measures are closed under conical combinationConical combination

Given a finite number of vectors x_1, x_2, \dots, x_n\, in a real vector space, a conical combination or a conical sum of these vectors is a vector of the formwhere the real numbers \alpha_i\, satisfy \alpha_i\ge 0...

but not general linear combination

Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

, while signed measures are the linear closure of positive measures.

Another generalization is the finitely additive measure, which are sometimes called contents. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first, but proved to be not so useful. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of L

^{∞}Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

and the Stone–Čech compactification

Stone–Cech compactification

In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX...

. All these are linked in one way or another to the axiom of choice.

A charge

Signed measure

In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Some authors may call it a charge, by analogy with electric charge, which is a familiar distribution that takes on positive and negative values.-Definition:There are two slightly...

is a generalization in both directions: it is a finitely additive, signed measure.