In mathematics the concept of a measure generalizes notions such as "length", "area", and "volume" (but not all of its applications have to do with physical sizes). Informally, given some base set, a "measure" is any consistent assignment of "sizes" to (some of) the subsets of the base set. Depending on the application, the "size" of a subset may be interpreted as (for example) its physical size, the amount of something that lies within the subset, or the probability that some random process will yield a result within the subset. The main use of measures is to define general concepts of integration over domains with more complex structure than intervals of the real line. Such integrals are used extensively in probability theory, and in much of mathematical analysis. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... Analysis has its beginnings in the rigorous formulation of calculus. ...

It is often not possible or desirable to assign a size to all subsets of the base set, so a measure does not have to do so. There are certain consistency conditions that govern which combinations of subsets it is allowed for a measure to assign sizes to; these conditions are encapsulated in the auxiliary concept of a σ-algebra. In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ...

Measure theory is that branch of real analysis which investigates σ-algebras, measures, measurable functions and integrals. Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ... The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically...

Informally, a measure maps sets to non-negative real numbers, with larger sets being mapped to bigger numbers.

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Definition

Formally, a measure μ is a function defined on a σ-algebra Σ over a set X and taking values in the extended interval [0,∞] such that the following properties are satisfied: Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...

• The empty set has measure zero:

.

• Countable additivity or σ-additivity: if ... is a countable sequence of pairwise disjoint sets in Σ, the measure of the union of all the 's is equal to the sum of the measures of each :

The triple (X,Σ,μ) is then called a measure space, and the members of Σ are called measurable sets. The empty set is the set containing no elements. ... Let μ be a measure on a sigma algebra Σ of subsets of a set X. An element A in Σ is said to have measure zero if μ(A)=0. ... In mathematics, a measure is a function that assigns a number, e. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, two sets are said to be disjoint if they have no element in common. ... In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ...

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. In mathematics, the definition of the probability space is the foundation of probability theory. ...

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 authors. For more details see Radon measure. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Analysis is that branch of mathematics which deals with the real numbers, complex numbers, and their functions. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... In mathematics, a Radon measure on a Hausdorff topological space X is a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular. ... In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a Radon measure on a Hausdorff topological space X is a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular. ...

Properties

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

Monotonicity

μ is monotonic: If E₁ and E₂ are measurable sets with then . A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right). ...

Measures of infinite unions of measurable sets

μ is subadditive: If E₁, E₂, E₃, ... is a countable sequence of sets in Σ, not necessarily disjoint, then A function f(x) is subadditive if for all x and y in the domain of f. ... In mathematics the term countable set is used to describe the size of a set, e. ...

.

μ is continuous from below: If E₁, E₂, E₃, ... are measurable sets and E_n is a subset of E_{n + 1} for all n, then the union of the sets E_n is measurable, and In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

.

Measures of infinite intersections of measurable sets

μ is continuous from above: If E₁, E₂, E₃, ... are measurable sets and E_{n + 1} is a subset of E_n for all n, then the intersection of the sets E_n is measurable; furthermore, if at least one of the E_n has finite measure, then In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...

.

This property is false without the assumption that at least one of the E_n has finite measure. For instance, for each n ∈ N, let

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

Sigma-finite measures

Main article: Sigma-finite measure

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 union of sets with finite measure. In mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called finite, if μ(X) is a finite real number (rather than ∞). The measure μ is called σ-finite, if X is the countable union of measurable sets of 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 separability of topological spaces. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... The integers are commonly denoted by the above symbol. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... 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 the subsets elements if this is finite, and ∞ if the subset is infinite. ... In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ...

Completeness

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). In mathematics, the symmetric difference of two sets is the set of elements which are in one of either set, but not in both. ...

Examples

Some important measures are listed here.

• The counting measure is defined by μ(S) = number of elements in S.
• The Lebesgue measure is the unique complete translation-invariant measure on a σ-algebra containing the intervals in R such that μ([0,1]) = 1.
• Circular angle measure is invariant under rotation.
• The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure and has a similar uniqueness property.
• The Hausdorff measure which is a refinement of the Lebesgue measure to some 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 μ_a (confer Dirac delta function) is given by μ_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 measures include: Borel measure, Jordan measure, Ergodic measure, Euler measure, Gauss measure, Baire measure, Radon 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 the subsets elements if this is finite, and ∞ if the subset is infinite. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... ∠, the angle symbol. ... A sphere rotating around its axis. ... In 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. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, the Hausdorff dimension is an extended non-negative real number, that is in the closed infinite interval [0, &#8734;], associated to any metric space . ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In probability theory, the probability P of some event E, denoted , is defined in such a way that P satisfies the Kolmogorov axioms. ... The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ... In mathematics, characteristic function can refer to any of several distinct concepts: The most common and universal usage is as a synonym for indicator function, that is the function which for every subset A of X, has value 1 at points of A and 0 at points of X âˆ’ A... In mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing the intervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure b − a (where a < b). ... In mathematics, the Jordan measure is an extension of the notion of size (length, area, volume) to shapes more complicated than, for example, a triangle, disk, or parallelipiped. ... In mathematics, specifically in ergodic theory, an ergodic measure is a measure that satisfies the ergodic hypothesis for a given map of a measurable space into itself. ... In mathematics, a Radon measure on a Hausdorff topological space X is a measure on the σ-algebra of Borel sets of X that is locally finite and inner regular. ...

Non-measurable sets

Main article: Non-measurable set

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. In mathematics, a non-measurable set is a set whose structure is so complicated it sheds light on the very notion of length, area or volume. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... In mathematics, the Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable. ... 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 subsets A, B and C such that A, B, C and B ∪ C are all congruent. ... The Banach–Tarski paradox: A ball can be decomposed and reassembled into two balls the same size as the original. ...

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 numbers 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 mainly 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. In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, or more specifically in measure theory, a complex measure is a generalisation of the concept of measure by letting it have complex values. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, projection-valued measures are used to express results in spectral theory. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ...

Another generalization is the finitely additive measure. 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. In mathematical analysis, a Banach limit is a linear functional defined on the set of all bounded real-valued sequences such that for any real-valued sequences and , the following conditions are satisfied: (linearity); if , then ; , where is the shift operator defined by . ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... Wikipedia does not have an article with this exact name. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

The remarkable result in integral geometry known as Hadwiger's theorem states that the space of translation-invariant, finitely additive, not-necessarily-nonnegative set functions defined on finite unions of compact convex sets in consists (up to scalar multiples) of one "measure" that is "homogeneous of degree k" for each k=0,1,2,...,n, and linear combinations of those "measures". "Homogeneous of degree k" means that rescaling any set by any factor c > 0 multiplies the set's "measure" by c^k. The one that is homogeneous of degree n is the ordinary n-dimensional volume. The one that is homogeneous of degree n-1 is the "surface volume". The one that is homogeneous of degree 1 is a mysterious function called the "mean width", a misnomer. The one that is homogeneous of degree 0 is the Euler characteristic. In mathematics, the term integral geometry in is used in two ways, which, although related, imply different views of the content of the subject. ... In integral geometry (otherwise called geometric probability theory), Hadwigers theorem states that the space of measures (see below) defined on finite unions of compact convex sets in Rn consists of one measure that is homogeneous of degree k for each k = 0, 1, 2, ..., n, and linear combinations of... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... Look up Convex set in Wiktionary, the free dictionary. ... It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...

See also

Look up measurable in
Wiktionary, the free dictionary.

Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ... In mathematics, in particular in measure theory, an outer measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. ... In mathematics, let S be a σ-algebra and μ be a measure on S. Following the notation in the text by Halmos, referenced below. ... In mathematics, the Hausdorff dimension is an extended non-negative real number, that is in the closed infinite interval [0, &#8734;], associated to any metric space . ... In mathematics, given two measurable spaces and measures on them, one can obtain the product measurable space and the product measure on that space. ... In mathematics, a pushforward measure (also push forward or push-forward) is obtained by transferring (pushing forward) a measure from one measurable space to another using a measurable function. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... The integral of a positive function can be interpreted as the area under a curve. ... In measure theory, Carathéodorys extension theorem proves that for a given set Ω, you can always extend a σ-finite measure defined on R to the σ-algebra generated by R, where R is a ring included in the power set of Ω; moreover, the extension is unique. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ... Steinhaus theorem is a theorem in real analysis, first proved by H. Steinhaus, concerning the difference set of a set of positive measure. ... In mathematics, geometric measure theory (GMT) is the study of the geometric properties of the measures of sets (typically in Euclidean spaces), including such things as arc lengths and areas. ...

References

• R. G. Bartle, 1995. The Elements of Integration and Lebesgue Measure. Wiley Interscience.
• 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 0-471-317160-0 Second edition.
• D. H. Fremlin, 2000. Measure Theory. Torres Fremlin.
• Paul Halmos, 1950. Measure theory. Van Nostrand and Co.
• 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. Introduction to Measure and Integration. Addison Wesley.
• 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.
• Some useful Cambridge Tripos Notes on Probability and Measure Theory link