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[[Image:Measure illustration.png|right|thumb|Informally, a measure has the property of being [[monotone function|monotone]] in the sense that if ''A'' is a [[subset]] of ''B'', the measure of ''A'' is less than or equal to the measure of ''B''. Furthermore, the measure of the [[empty set]] is required to be 0.]]

In [[mathematical analysis]], a '''measure''' on a [[set (mathematics)|set]] is a systematic way to assign a number to each suitable [[subset]] of that set, intuitively interpreted as its size. 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 <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math>. For instance, the Lebesgue measure of the [[Interval (mathematics)|interval]] <math>\left[0, 1\right]</math> in the [[real line|real numbers]] is its length in the everyday sense of the word&thinsp;–&thinsp;specifically, 1.

Technically, a measure is a function that assigns a non-negative real number or [[Extended real number line|+∞]] to (certain) subsets of a set <math>X</math> (''see'' [[#Definition|Definition]] below). It must assign 0 to the [[empty set]] and be ([[countably]]) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a ''consistent'' size to ''each'' subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the [[counting measure]]. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called ''measurable'' subsets, which are required to form a [[Sigma-algebra|<math>\sigma</math>-algebra]]. This means that countable [[union (set theory)|unions]], countable [[intersection (set theory)|intersections]] and [[complement (set theory)|complements]] 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 complement. Indeed, their existence is a non-trivial consequence of the [[axiom of choice]].

Measure theory was developed in successive stages during the late 19<sup>th</sup> and early 20<sup>th</sup> 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]].

==Definition==
[[File:Countable additivity of a measure.svg|thumb|300px|Countable additivity of a measure <math>\mu</math>: The measure of a countable disjunctive union is the same as the sum of all measures of each subset.]]
Let <math>X</math> be a set and <math>\Sigma</math> a [[Sigma-algebra|<math>\sigma</math>-algebra]] over <math>X</math>. A [[function (mathematics)|function]] <math>\mu</math> from <math>\Sigma</math> to the [[extended real number line]] is called a '''measure''' if it satisfies the following properties:

*'''Non-negativity''':
:<math>\forall E \in \Sigma : \mu\!\left(E\right) \geq 0</math>.

*'''Null empty set''':
:<math>\mu\!\left(\varnothing\right) = 0</math>.

*'''Countable additivity''' (or [[sigma additivity|<math>\sigma</math>-additivity]]): For all [[countable]] collections <math>\left\{E_i\right\}_{i \in \mathbf{N}}</math> of pairwise [[disjoint sets]] in <math>\Sigma</math>:
:<math>\mu\Bigl(\bigcup_{i \in \mathbf{N}} E_i\Bigr) = \sum_{i \in \mathbf{N}} \mu\!\left(E_i\right)</math>.

One may require that at least one set <math>E</math> has finite measure. Then the null set automatically has measure zero because of countable additivity, because <math>\mu\!\left(E\right)=\mu\!\left(E \cup \varnothing\right) = \mu\!\left(E\right) + \mu\!\left(\varnothing\right)</math>, so <math>\mu\!\left(\varnothing\right) = \mu\!\left(E\right) - \mu\!\left(E\right) = 0</math>.

If only the second and third conditions of the definition of measure above are met, and <math>\mu</math> takes on at most one of the values <math>\pm\infty</math>, then <math>\mu</math> is called a '''[[signed measure]]'''.

{{anchor|Measurable space}}
The pair <math>\left(X, \Sigma\right)</math> is called a '''measurable space''', the members of <math>\Sigma</math> are called '''measurable sets'''. If <math>\left(X, \Sigma_X\right)</math> and <math>\left(Y, \Sigma_Y\right)</math> are two measurable spaces, then a function <math>f\colon X \to Y</math> is called '''measurable''' if for every <math>Y</math>-measurable set <math>B \in \Sigma_Y</math>, the [[Image (mathematics)#Inverse image|inverse image]] is <math>X</math>-measurable&thinsp;–&thinsp;i.e.: <math>f^{\left(-1\right)}\!\left(B\right) \in \Sigma_X</math>. The [[Function composition|composition]] of measurable functions is measurable, making the measurable spaces and measurable functions a [[Category (mathematics)|category]], with the measurable spaces as objects and the set of measurable functions as arrows.

{{anchor|Measure spaces}}
A [[tuple|triple]] <math>\left(X, \Sigma, \mu\right)</math> is called a '''{{visible anchor|measure space}}'''. A [[probability measure]] is a measure with total measure one&thinsp;–&thinsp;i.e. <math>\mu\!\left(X\right) = 1</math>&thinsp;–&thinsp; 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 (mathematics)|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 [[support (mathematics)#Compact support|compact support]].{{Clarify|date=March 2013}} This approach is taken by [[Nicolas Bourbaki|Bourbaki]] (2004) and a number of other sources. For more details, see the article on [[Radon measure]]s.

==Properties==
Several further properties can be derived from the definition of a countably additive measure.

===Monotonicity===
A measure μ is [[monotonic function|monotonic]]: If ''E''<sub>1</sub> and ''E''<sub>2</sub> are measurable sets with ''E''<sub>1</sub>&nbsp;⊆ ''E''<sub>2</sub> then
:<math>\mu(E_1) \leq \mu(E_2).</math>

===Measures of infinite unions of measurable sets===
A measure μ is countably [[subadditivity|subadditive]]:  For any [[countable]] [[Sequence (mathematics)|sequence]] {{math|''E''<sub>1</sub>, ''E''<sub>2</sub>, ''E''<sub>3</sub>,…}} of sets {{math|''E''<sub><var>n</var></sub>}} in {{math|&Sigma;}} (not necessarily disjoint):
:<math>\mu\left( \bigcup_{i=1}^\infty E_i\right) \le \sum_{i=1}^\infty \mu(E_i).</math>

A measure μ is continuous from below: If {{math|''E''<sub>1</sub>, ''E''<sub>2</sub>, ''E''<sub>3</sub>,…}} are measurable sets and {{math|''E''<sub>''n''</sub>}} is a subset of {{math|''E''<sub><var>n</var> + 1</sub>}} for all {{math|<var>n</var>}}, then the [[Union (set theory)|union]] of the sets {{math|''E''<sub><var>n</var></sub>}} is measurable, and
:<math> \mu\left(\bigcup_{i=1}^\infty E_i\right) = \lim_{i\to\infty}  \mu(E_i).</math>

===Measures of infinite intersections of measurable sets===
A measure <math>\mu</math> is continuous from above: If <math>E_1, E_2, E_3, \dots</math>, are measurable sets and <math>\forall n, E_{n+1} \subset E_{n}</math>, then the [[Intersection (set theory)|intersection]] of the sets <math>E_{n}</math> is measurable; furthermore, if at least one of the <math>E_{n}</math> has finite measure, then

:<math> \mu\left(\bigcap_{i=1}^\infty E_i\right) = \lim_{i\to\infty} \mu(E_i).</math>

This property is false without the assumption that at least one of the <math>E_{n}</math> has finite measure. For instance, for each <math>n \in \mathbb{N}</math>, let

:<math> E_{n} = [n, \infty) \subseteq \mathbb{R} </math>

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

==Sigma-finite measures==
{{Main|Sigma-finite measure}}

A measure space (''X'', Σ, μ) is called finite if μ(''X'') is a finite real number (rather than ∞). Nonzero finite measures are analogous to probability measures in the sense that any finite measure <math>\mu</math> is proportional to the probability measure <math>\frac{1}{\mu(X)}\mu</math>. A measure <math>\mu</math> is called ''σ-finite'' if ''X'' can be decomposed into a countable union of measurable sets of finite measure. Analogously, a set in a measure space is said to have a ''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 interval]]s [''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 space|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'.

== Completeness ==
{{Main|Complete 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'').

== 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<sub>i</sub>'', <math>i\in I</math> define:
:<math>\sum_{i\in I} r_i=\sup\left\lbrace\sum_{i\in J} r_i : |J|<\aleph_0, J\subseteq I\right\rbrace.</math>
That is, we define the sum of the <math>r_i</math> to be the supremum of all the sums of finitely many of them.

A measure <math>\mu</math> on <math>\Sigma</math> is <math>\kappa</math>-additive if for any <math>\lambda<\kappa</math> and any family <math>X_\alpha</math>, <math>\alpha<\lambda</math> the following hold:
#<math>\bigcup_{\alpha\in\lambda} X_\alpha \in \Sigma</math>
#<math>\mu\left(\bigcup_{\alpha\in\lambda} X_\alpha\right)=\sum_{\alpha\in\lambda}\mu\left(X_\alpha\right).</math>
Note that the second condition is equivalent to the statement that the [[Ideal (set theory)|ideal]] of null sets is <math>\kappa</math>-complete.

==Examples==

Some important measures are listed here.

* The [[counting measure]] is defined by μ(''S'') = number of elements in ''S''.
* The [[Lebesgue measure]] on '''R''' is a complete [[translational invariance|translation-invariant]] measure on a ''σ''-algebra containing the [[interval (mathematics)|interval]]s in '''R''' such that μ([0,1]) = 1; and every other measure with these properties extends Lebesgue measure.
* Circular [[angle]] measure is invariant under [[rotation]], and [[hyperbolic angle]] measure is invariant under [[squeeze mapping]].
* The [[Haar measure]] for a [[Locally compact space|locally compact]] [[topological group]] is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties.
* The [[Hausdorff measure]] is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets.
* Every [[probability space]] gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the [[unit interval]] [0,1]). Such a measure is called a ''probability measure''. See [[probability axioms]].
* The [[Dirac measure]] δ<sub>''a''</sub> (cf. [[Dirac delta function]]) is given by δ<sub>''a''</sub>(''S'') = χ<sub>''S''</sub>(a), where χ<sub>''S''</sub> is the [[Indicator function|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]], [[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 quantity|conserved]] (see [[conservation law]] for a list of these) or not. Negative values lead to signed measures, see "generalizations" below.

[[Liouville's theorem (Hamiltonian)#Symplectic geometry|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]].

==Non-measurable sets==
{{Main|Non-measurable set}}

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]].

== 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 function]] 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.{{Citation needed|date=September 2011}} 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 '''positive measure''' is used. Positive measures are closed under [[conical combination]] 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 [[Content (measure theory)|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. It turns out that in general, finitely additive measures are connected with notions such as [[Banach limit]]s, the dual of [[lp space|''L''<sup>∞</sup>]] and the [[Stone–Čech compactification]]. All these are linked in one way or another to the [[axiom of choice]].

A [[signed measure|charge]] is a generalization in both directions: it is a finitely additive, signed measure.

==See also==
<div style="-moz-column-count:3; column-count:3;">
* [[Abelian von Neumann algebra]]
* [[Almost everywhere]]
* [[Carathéodory's extension theorem]]
* [[Fubini's theorem]]
* [[Fuzzy measure theory]]
* [[Geometric measure theory]]
* [[Hausdorff measure]]
* [[Inner measure]]
* [[Lebesgue integration]]
* [[Lebesgue measure]]
* [[Lifting theory]]
* [[Measurable function]]
* [[Outer measure]]
* [[Product measure]]
* [[Pushforward measure]]
* [[Vector measure]]
* [[Volume form]]
* [[Measurable cardinal]]
</div>

== References ==
* [[Robert G. Bartle]] (1995) ''The Elements of Integration and Lebesgue Measure'', Wiley Interscience.
* {{citation | last=Bauer|first=H.|title=Measure and Integration Theory|year=2001|publisher=de Gruyter|location=Berlin|isbn=978-3110167191}}
* {{citation | last=Bear|first=H.S.|title=A Primer of Lebesgue Integration|year=2001|publisher=Academic Press|location=San Diego|isbn=978-0120839711}}
* {{citation | last=Bogachev|first=V. I.|title=Measure theory|year=2006|publisher=Springer|location=Berlin|isbn=978-3540345138}}
* {{citation | last=Bourbaki| first=Nicolas | title=Integration I |  year=2004 | publisher=[[Springer Verlag]] | isbn=3-540-41129-1}} Chapter III.
* R. M. Dudley, 2002. ''Real Analysis and Probability''. Cambridge University Press.
* {{citation | last=Folland | first=Gerald B.| title=Real Analysis: Modern Techniques and Their Applications | year = 1999 | publisher = John Wiley and Sons | isbn=0471317160 }} Second edition.
* D. H. Fremlin, 2000. ''[http://www.essex.ac.uk/maths/people/fremlin/mt.htm Measure Theory]''. Torres Fremlin.
* {{citation|first=Paul|last= Halmos|year= 1950|title=Measure theory|publisher= Van Nostrand and Co.}}
* {{citation | last=Jech| first=Thomas | title=Set Theory: The Third Millennium Edition, Revised and Expanded |  year=2003 | publisher=[[Springer Verlag]] | isbn=3-540-44085-2}}
* [[R. Duncan Luce]] and Louis Narens (1987). "measurement, theory of," ''The [[New Palgrave: A Dictionary of Economics]]'', v. 3, pp.&nbsp;428&ndash;32.
* M. E. Munroe, 1953. ''Introduction to Measure and Integration''. Addison Wesley.
* {{Citation|author=K. P. S. Bhaskara Rao and M. Bhaskara Rao|title=Theory of Charges: A Study of Finitely Additive Measures| publisher=Academic Press|location=London|year=1983|pages=x + 315|isbn=0-12-095780-9}}
* Shilov, G. E., and Gurevich, B. L., 1978. ''Integral, Measure, and Derivative: A Unified Approach'', Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8. Emphasizes the [[Daniell integral]].
* {{citation | last = Teschl| first = Gerald| authorlink = Gerald Teschl| title = Topics in Real and Functional Analysis| url = http://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html|publisher = (lecture notes)}}
* [[Terence Tao]], 2011. ''An Introduction to Measure Theory''.  American Mathematical Society.
* [[Nik Weaver]], 2013. ''Measure Theory and Functional Analysis''.  World Scientific Publishing.

==External links==
{{Wiktionary|measurable}}
*{{springer|title=Measure|id=p/m063240}}
* [http://www.ee.washington.edu/techsite/papers/documents/UWEETR-2006-0008.pdf Tutorial: Measure Theory for Dummies]

[[Category:Mathematical analysis]]
[[Category:Measure theory| ]]
[[Category:Measures (measure theory)| ]]