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In mathematics, a **measure** on a set is a systematic way to assign a number to subsets of a set, intuitively interpreted as the size of the subset. Those sets which can be associated with such a number, we call **measurable sets**. 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. This assigns the usual length, area, or volume to certain subsets of the given Euclidean space. For instance, the Lebesgue measure of an interval of real numbers is its usual length, but also assigns numbers to other kinds of sets in a way that is consistent with the lengths of intervals.

- Definition
- Instances
- Basic properties
- Monotonicity
- Measure of countable unions and intersections
- Other properties
- Completeness
- Additivity
- Sigma-finite measures
- s-finite measures
- Non-measurable sets
- Generalizations
- See also
- References
- Bibliography
- External links

Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see * § Definition *, below). A measure must further be countably additive: if a 'large' subset can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets that are measurable, then the 'large' subset is *measurable*, and its measure is the sum (possibly infinite) of the measures of the "smaller" subsets.

In general, if one wants to associate a *consistent* size to *all* subsets 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 σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets 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.^{ [1] } 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 integrals 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.

Let X be a set and Σ a σ-algebra over X. A function μ from Σ to the extended real number line is called a **measure** if it satisfies the following properties:

**Non-negativity**: For all*E*in Σ, we have*μ*(*E*) ≥ 0.**Null empty set**: .**Countable additivity**(or σ-additivity): For all countable collections of pairwise disjoint sets in Σ,

If at least one set has finite measure, then the requirement that is met automatically. Indeed, by countable additivity,

and therefore

If the condition of non-negativity is omitted but the second and third of these conditions are met, and μ takes on at most one of the values ±∞, then μ is called a ** signed measure **.

The pair (*X*, Σ) is called a measurable space, the members of Σ are called **measurable sets**. If and are two measurable spaces, then a function is called **measurable** if for every *Y*-measurable set , the inverse image is X-measurable – i.e.: . In this setup, the composition of measurable functions is measurable, making the measurable spaces and measurable functions a category, with the measurable spaces as objects and the set of measurable functions as arrows. See also * Measurable function § Term usage variations * about another setup.

A triple (*X*, Σ, *μ*) is called a measure space. A probability measure is a measure with total measure one – i.e. *μ*(*X*) = 1. A probability space is a measure space with a probability measure.

For measure spaces that are also topological spaces 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 measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other sources. For more details, see the article on Radon measures.

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 translation-invariant measure on a*σ*-algebra containing the intervals 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 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*_{a}(cf. Dirac delta function) is given by*δ*_{a}(*S*) =*χ*_{S}(a), where*χ*_{S}is the indicator 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, Young measure, and Loeb 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.

Let μ be a measure.

If *E*_{1} and *E*_{2} are measurable sets with *E*_{1} ⊆ *E*_{2} then

For any countable sequence *E*_{1}, *E*_{2}, *E*_{3}, ... of (not necessarily disjoint) measurable sets *E _{n}* in Σ:

If *E*_{1}, *E*_{2}, *E*_{3}, ... are measurable sets and for all *n*, then the union of the sets *E _{n}* is measurable, and

If *E*_{1}, *E*_{2}, *E*_{3}, ... are measurable sets and, for all *n*, then the intersection of the sets *E _{n}* is measurable; furthermore, if at least one of the

This property is false without the assumption that at least one of the *E _{n}* has finite measure. For instance, for each

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*).

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

That is, we define the sum of the to be the supremum of all the sums of finitely many of them.

A measure on is -additive if for any and any family of disjoint sets the following hold:

Note that the second condition is equivalent to the statement that the ideal of null sets is -complete.

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 μ is proportional to the probability measure . A measure μ 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 numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals [*k*, *k*+1] for all integers *k*; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers 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 measure is said to be s-finite if it is a countable sum of bounded measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes.

If the axiom of choice is assumed to be true, it can be proved that 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.

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 numbers is called a * complex measure *. Measures that take values in Banach spaces have been studied extensively.^{ [2] } 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*, also known as a content. 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 limits, the dual of *L*^{∞} and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice. Contents remain useful in certain technical problems in geometric measure theory; this is the theory of Banach measures.

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

- Abelian von Neumann algebra
- Almost everywhere
- Carathéodory's extension theorem
- Content (measure theory)
- Fubini's theorem
- Fatou's lemma
- Fuzzy measure theory
- Geometric measure theory
- Hausdorff measure
- Inner measure
- Lebesgue integration
- Lebesgue measure
- Lorentz space
- Lifting theory
- Measurable cardinal
- Measurable function
- Minkowski content
- Outer measure
- Product measure
- Pushforward measure
- Regular measure
- Vector measure
- Valuation (measure theory)
- Volume form

In mathematics, specifically in measure theory, a **Borel measure** on a topological space is a measure that is defined on all open sets. Some authors require additional restrictions on the measure, as described below.

In mathematical analysis and in probability theory, a **σ-algebra** on a set *X* is a collection of subsets of *X* that includes *X* itself, is closed under complement, and is closed under countable unions.

In measure theory, a property holds **almost everywhere** if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of *almost surely* in probability theory.

In calculus, **absolute continuity** is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: *absolute continuity of functions* and *absolute continuity of measures.* These two notions are generalized in different directions. The usual derivative of a function is related to the *Radon–Nikodym derivative*, or *density*, of a measure.

In mathematical analysis **Fubini's theorem**, introduced by Guido Fubini in 1907, is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value.

In mathematics, the **Radon–Nikodym theorem** is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A *measure* is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volume, where the subsets are sets of points; or the probability of an event, which is a subset of possible outcomes within a wider probability space.

In measure theory, Lebesgue's **dominated convergence theorem** provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the *L*^{1} norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration.

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 measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures. Carathéodory's work on outer measures found many applications in measure-theoretic set theory, and was used in an essential way by Hausdorff to define a dimension-like metric invariant now called Hausdorff dimension. Outer measures are commonly used in the field of geometric measure theory.

In mathematics, the **ba space** of an algebra of sets is the Banach space consisting of all bounded and finitely additive signed measures on . The norm is defined as the variation, that is ,

In mathematics, an **additive set function** is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, . If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of *k* disjoint sets equals the sum of its values on the sets. Therefore, an additive set funciton is also called a **finitely-additive set function**. However, a finitely-additive set function might not have the additivity property for a union of an *infinite* number of sets. A **σ-additive set function** is a function that has the additivity property even for infinitely many sets, that is, .

In mathematics, more precisely in measure theory, an **atom** is a measurable set which has positive measure and contains no set of smaller positive measure. A measure which has no atoms is called **non-atomic** or **atomless**.

In mathematics, a **regular measure** on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets.

In mathematics, the **Bochner integral**, named for Salomon Bochner, extends the definition of Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions.

In measure theory, **Carathéodory's extension theorem** states that any pre-measure defined on a given ring *R* of subsets of a given set *Ω* can be extended to a measure on the σ-algebra generated by *R*, and this extension is unique if the pre-measure is σ-finite. Consequently, any pre-measure on a ring containing all intervals of real numbers can be extended to the Borel algebra of the set of real numbers. This is an extremely powerful result of measure theory, and leads, for example, to the Lebesgue measure.

In mathematics, a positive measure *μ* defined on a *σ*-algebra Σ of subsets of a set *X* is called a finite measure if *μ*(*X*) is a finite real number, and a set *A* in Σ is of finite measure if *μ*(*A*) < ∞*.* The measure *μ* is called **σ-finite** if *X* is the countable union of measurable sets with finite measure. A set in a measure space is said to have ** σ-finite measure** if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.e. all finite measures are σ-finite but there are (many) σ-finite measures that are not finite.

In mathematics, more precisely in measure theory, a measure on the real line is called a **discrete measure** if it is concentrated on an at most countable set. Note that the support need not be a discrete set. Geometrically, a discrete measure is a collection of point masses.

In mathematics, a **content** is a set function that is like a measure, but a content must only be finitely additive, whereas a measure must be countably additive. A content is a real function defined on a collection of subsets such that

In probability theory, a **standard probability space**, also called **Lebesgue–Rokhlin probability space** or just **Lebesgue space** is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms.

In mathematics, **lifting theory** was first introduced by John von Neumann in a pioneering paper from 1931, in which he answered a question raised by Alfréd Haar. The theory was further developed by Dorothy Maharam (1958) and by Alexandra Ionescu Tulcea and Cassius Ionescu Tulcea (1961). Lifting theory was motivated to a large extent by its striking applications. Its development up to 1969 was described in a monograph of the Ionescu Tulceas. Lifting theory continued to develop since then, yielding new results and applications.

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 *x*-axis. The **Lebesgue integral** extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.

- ↑ Halmos, Paul (1950),
*Measure theory*, Van Nostrand and Co. - ↑ Rao, M. M. (2012),
*Random and Vector Measures*, Series on Multivariate Analysis,**9**, World Scientific, ISBN 978-981-4350-81-5, MR 2840012 .

- Robert G. Bartle (1995)
*The Elements of Integration and Lebesgue Measure*, Wiley Interscience. - Bauer, H. (2001),
*Measure and Integration Theory*, Berlin: de Gruyter, ISBN 978-3110167191 - Bear, H.S. (2001),
*A Primer of Lebesgue Integration*, San Diego: Academic Press, ISBN 978-0120839711 - Bogachev, V. I. (2006),
*Measure theory*, Berlin: Springer, ISBN 978-3540345138 - Bourbaki, Nicolas (2004),
*Integration I*, Springer Verlag, ISBN 3-540-41129-1 Chapter III. - R. M. Dudley, 2002.
*Real Analysis and Probability*. Cambridge University Press. - Folland, Gerald B. (1999),
*Real Analysis: Modern Techniques and Their Applications*, John Wiley and Sons, ISBN 0471317160 Second edition. - Federer, Herbert. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969 xiv+676 pp.
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*Measure Theory*. Torres Fremlin. - Jech, Thomas (2003),
*Set Theory: The Third Millennium Edition, Revised and Expanded*, 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. 428–32. - M. E. Munroe, 1953.
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*Integral, Measure, and Derivative: A Unified Approach*, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8. Emphasizes the Daniell integral. - Teschl, Gerald,
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