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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the concept of a measure is a generalization and formalization of geometrical measures ( length,
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
,
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
) and other common notions, such as
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
and
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, integration theory, and can be generalized to assume negative values, as with
electrical charge Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described ...
. Far-reaching generalizations (such as
spectral measure In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his disse ...
s and
projection-valued measure In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are ...
s) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to
ancient Greece Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean Sea, Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, classical antiquity ( AD 600), th ...
, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biography Borel was ...
,
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
,
Nikolai Luzin Nikolai Nikolaevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlaɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 January 1950) was a Soviet/Ru ...
,
Johann Radon Johann Karl August Radon (; 16 December 1887 – 25 May 1956) was an Austrian mathematician. His doctoral dissertation was on the calculus of variations (in 1910, at the University of Vienna). Life RadonBrigitte Bukovics: ''Biography of Johan ...
,
Constantin Carathéodory Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
, and
Maurice Fréchet Maurice may refer to: People *Saint Maurice (died 287), Roman legionary and Christian martyr *Maurice (emperor) or Flavius Mauricius Tiberius Augustus (539–602), Byzantine emperor * Maurice (bishop of London) (died 1107), Lord Chancellor and L ...
, among others.


Definition

Let X be a set and \Sigma a \sigma-algebra over X. A
set function In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R an ...
\mu from \Sigma to the extended real number line is called a measure if it satisfies the following properties: *Non-negativity: For all E in \Sigma, we have \mu(E) \geq 0. *Null empty set: \mu(\varnothing) = 0. *Countable additivity (or \sigma-additivity): For all
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
collections \_^\infty of pairwise disjoint sets in Σ,\mu\left(\bigcup_^\infty E_k\right)=\sum_^\infty \mu(E_k). If at least one set E has finite measure, then the requirement that \mu(\varnothing) = 0 is met automatically. Indeed, by countable additivity, \mu(E)=\mu(E \cup \varnothing) = \mu(E) + \mu(\varnothing), and therefore \mu(\varnothing)=0. If the condition of non-negativity is omitted but the second and third of these conditions are met, and \mu takes on at most one of the values \pm \infty, then \mu is called a ''
signed measure In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values. Definition There are two slightly different concepts of a signed measure, depending on whether or not ...
''. The pair (X, \Sigma) is called a ''
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the ...
'', and the members of \Sigma are called measurable sets. A
triple Triple is used in several contexts to mean "threefold" or a " treble": Sports * Triple (baseball), a three-base hit * A basketball three-point field goal * A figure skating jump with three rotations * In bowling terms, three strikes in a row * ...
(X, \Sigma, \mu) is called a '' measure space''. A probability measure is a measure with total measure one – that is, \mu(X) = 1. A
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
is a measure space with a probability measure. For measure spaces that are also
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
(and in many cases also in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
) are
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
s. Radon measures have an alternative definition in terms of linear functionals on the
locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
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 measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
s.


Instances

Some important measures are listed here. * The
counting measure In mathematics, specifically measure theory, 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 has finitely many elements, and infinity ...
is defined by \mu(S) = number of elements in S. * The Lebesgue measure on \R is a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
translation-invariant measure on a ''σ''-algebra containing the intervals in \R such that \mu( , 1 = 1; and every other measure with these properties extends Lebesgue measure. * Circular
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
measure is invariant under rotation, and
hyperbolic angle In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic function ...
measure is invariant under
squeeze mapping In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping. For a fixed positive real number , th ...
. * The Haar measure for a locally compact
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties. * The
Hausdorff measure In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that as ...
is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets. * Every
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the
unit interval In 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. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
, 1. Such a measure is called a ''probability measure''. See
probability axioms The Kolmogorov axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probabili ...
. * The
Dirac measure In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. ...
''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 In mathematics, the Peano–Jordan measure (also known as the Jordan content) is an extension of the notion of size (length, area, volume) to shapes more complicated than, for example, a triangle, disk, or parallelepiped. It turns out that for a ...
, ergodic measure,
Gaussian measure In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are nam ...
,
Baire measure In mathematics, a Baire measure is a measure on the σ-algebra of Baire sets of a topological space whose value on every compact Baire set is finite. In compact metric spaces the Borel sets and the Baire sets are the same, so Baire measures are the ...
,
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
, Young measure, and
Loeb measure In mathematics, a Loeb space is a type of measure space introduced by using nonstandard analysis. Construction Loeb's construction starts with a finitely additive map \nu from an internal algebra \mathcal A of sets to the nonstandard reals. Def ...
. In physics an example of a measure is spatial distribution of
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
(see for example,
gravity potential In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric po ...
), or another non-negative
extensive property Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...
, conserved (see conservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below. *
Liouville measure In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sym ...
, known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics. *
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 a generalization of the canonical ensemble to infinite systems. Th ...
is widely used in statistical mechanics, often under the name
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat ...
.


Basic properties

Let \mu be a measure.


Monotonicity

If E_1 and E_2 are measurable sets with E_1 \subseteq E_2 then \mu(E_1) \leq \mu(E_2).


Measure of countable unions and intersections


Subadditivity

For any
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
E_1, E_2, E_3, \ldots of (not necessarily disjoint) measurable sets E_n in \Sigma: \mu\left( \bigcup_^\infty E_i\right) \leq \sum_^\infty \mu(E_i).


Continuity from below

If E_1, E_2, E_3, \ldots are measurable sets that are increasing (meaning that E_1 \subseteq E_2 \subseteq E_3 \subseteq \ldots) then the
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
of the sets E_n is measurable and \mu\left(\bigcup_^\infty E_i\right) ~=~ \lim_ \mu(E_i) = \sup_ \mu(E_i).


Continuity from above

If E_1, E_2, E_3, \ldots are measurable sets that are decreasing (meaning that E_1 \supseteq E_2 \supseteq E_3 \supseteq \ldots) then the intersection of the sets E_n is measurable; furthermore, if at least one of the E_n has finite measure then \mu\left(\bigcap_^\infty E_i\right) = \lim_ \mu(E_i) = \inf_ \mu(E_i). This property is false without the assumption that at least one of the E_n has finite measure. For instance, for each n \in \N, let E_n = [n, \infty) \subseteq \R, which all have infinite Lebesgue measure, but the intersection is empty.


Other properties


Completeness

A measurable set X is called a ''null set'' if \mu(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 In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. Th ...
of X and Y is contained in a null set. One defines \mu(Y) to equal \mu(X).


μ = μ (a.e.)

If f:X\to ,+\infty/math> is (\Sigma,( ,+\infty)-measurable, then \mu\ = \mu\ for almost all t \in X. This property is used in connection with
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
.


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,i\in I define: \sum_ r_i=\sup\left\lbrace\sum_ r_i : , J, <\aleph_0, J\subseteq I\right\rbrace. That is, we define the sum of the r_i to be the supremum of all the sums of finitely many of them. A measure \mu on \Sigma is \kappa-additive if for any \lambda<\kappa and any family of disjoint sets X_\alpha,\alpha<\lambda the following hold: \bigcup_ X_\alpha \in \Sigma \mu\left(\bigcup_ X_\alpha\right) = \sum_\mu\left(X_\alpha\right). Note that the second condition is equivalent to the statement that the
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
of null sets is \kappa-complete.


Sigma-finite measures

A measure space (X, \Sigma, \mu) is called finite if \mu(X) is a finite real number (rather than \infty). Nonzero finite measures are analogous to probability measures in the sense that any finite measure \mu is proportional to the probability measure \frac\mu. A measure \mu 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 In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s with the standard Lebesgue measure are σ-finite but not finite. Consider the
closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
s , k+1/math> for all
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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 In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s with the
counting measure In mathematics, specifically measure theory, 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 has finitely many elements, and infinity ...
, 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'.


Strictly localizable measures


Semifinite measures

Let X be a set, let be a sigma-algebra on X, and let \mu be a measure on . We say \mu is semifinite to mean that for all A\in\mu^\text\, (A)\cap\mu^\text(\R_)\ne\emptyset. Semifinite measures generalize sigma-finite measures, in such a way that some big theorems of measure theory that hold for sigma-finite but not arbitrary measures can be extended with little modification to hold for semifinite measures. (To-do: add examples of such theorems; cf. the talk page.)


Basic examples

* Every sigma-finite measure is semifinite. * Assume =(X), let f:X\to ,+\infty and assume \mu(A)=\sum_f(a) for all A\subseteq X. ** We have that \mu is sigma-finite if and only if f(x)<+\infty for all x\in X and f^\text(\R_) is countable. We have that \mu is semifinite if and only if f(x)<+\infty for all x\in X. ** Taking f=X\times\ above (so that \mu is counting measure on (X)), we see that counting measure on (X) is *** sigma-finite if and only if X is countable; and *** semifinite (without regard to whether X is countable). (Thus, counting measure, on the power set (X) of an arbitrary uncountable set X, gives an example of a semifinite measure that is not sigma-finite.) * Let d be a complete, separable metric on X, let be the Borel sigma-algebra induced by d, and let s\in\R_. Then the
Hausdorff measure In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that as ...
^s, is semifinite. * Let d be a complete, separable metric on X, let be the Borel sigma-algebra induced by d, and let s\in\R_. Then the packing measure ^s, is semifinite.


Involved example

The zero measure is sigma-finite and thus semifinite. In addition, the zero measure is clearly less than or equal to \mu. It can be shown there is a greatest measure with these two properties: We say the semifinite part of \mu to mean the semifinite measure \mu_\text defined in the above theorem. We give some nice, explicit formulas, which some authors may take as definition, for the semifinite part: * \mu_\text=(\sup\)_. * \mu_\text=(\sup\)_\}. * \mu_\text=\mu, _\cup\\times\\cup\\times\. Since \mu_\text is semifinite, it follows that if \mu=\mu_\text then \mu is semifinite. It is also evident that if \mu is semifinite then \mu=\mu_\text.


Non-examples

Every ''0-\infty measure'' that is not the zero measure is not semifinite. (Here, we say ''0-\infty measure'' to mean a measure whose range lies in \: (\forall A\in)(\mu(A)\in\).) Below we give examples of 0-\infty measures that are not zero measures. * Let X be nonempty, let be a \sigma-algebra on X, let f:X\to\ be not the zero function, and let \mu=(\sum_f(x))_. It can be shown that \mu is a measure. ** \mu=\\cup(\setminus\)\times\. *** X=\, =\, \mu=\. * Let X be uncountable, let be a \sigma-algebra on X, let =\ be the countable elements of , and let \mu=\times\\cup(\setminus)\times\. It can be shown that \mu is a measure.


Involved non-example

We say the \mathbf part of \mu to mean the measure \mu_ defined in the above theorem. Here is an explicit formula for \mu_: \mu_=(\sup\)_.


Results regarding semifinite measures

* Let \mathbb F be \R or \C, and let T:L_\mathbb^\infty(\mu)\to\left(L_\mathbb^1(\mu)\right)^*:g\mapsto T_g=\left(\int fgd\mu\right)_. Then \mu is semifinite if and only if T is injective. (This result has import in the study of the dual space of L^1=L_\mathbb^1(\mu).) * Let \mathbb F be \R or \C, and let be the topology of convergence in measure on L_\mathbb^0(\mu). Then \mu is semifinite if and only if is Hausdorff. * (Johnson) Let X be a set, let be a sigma-algebra on X, let \mu be a measure on , let Y be a set, let be a sigma-algebra on Y, and let \nu be a measure on . If \mu,\nu are both not a 0-\infty measure, then both \mu and \nu are semifinite if and only if (\mu\times_\text\nu)(A\times B)=\mu(A)\nu(B) for all A\in and B\in. (Here, \mu\times_\text\nu is the measure defined in Theorem 39.1 in Berberian '65.)


Localizable measures

Localizable measures are a special case of semifinite measures and a generalization of sigma-finite measures. Let X be a set, let be a sigma-algebra on X, and let \mu be a measure on . * Let \mathbb F be \R or \C, and let T : L_\mathbb^\infty(\mu) \to \left(L_\mathbb^1(\mu)\right)^* : g \mapsto T_g = \left(\int fgd\mu\right)_. Then \mu is localizable if and only if T is bijective (if and only if L_\mathbb^\infty(\mu) "is" L_\mathbb^1(\mu)^*).


s-finite measures

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.


Non-measurable sets

If the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
is assumed to be true, it can be proved that not all subsets of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
are
Lebesgue measurable In measure theory, a branch of mathematics, 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 wit ...
; examples of such sets include the
Vitali set In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905. The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vita ...
, and the non-measurable sets postulated by the
Hausdorff paradox The Hausdorff paradox is a paradox in mathematics named after Felix Hausdorff. It involves the sphere (a 3-dimensional sphere in ). It states that if a certain countable subset is removed from , then the remainder can be divided into three disjoin ...
and the
Banach–Tarski paradox The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be p ...
.


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 In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R an ...
with values in the (signed) real numbers is called a ''
signed measure In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values. Definition There are two slightly different concepts of a signed measure, depending on whether or not ...
'', while such a function with values in the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s is called a ''
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 (length, area, volume) is a complex number. Definition Formal ...
''. Observe, however, that complex measure is necessarily of finite variation, hence complex measures include finite signed measures but not, for example, the Lebesgue measure. Measures that take values in
Banach spaces In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
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 In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are ...
''; these are used in
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 (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
for the
spectral theorem In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
. 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 Given a finite number of vectors x_1, x_2, \dots, x_n in a real vector space, a conical combination, conical sum, or weighted sum''Convex Analysis and Minimization Algorithms'' by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993, pp. 101, 102/ ...
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 Content or contents may refer to: Media * Content (media), information or experience provided to audience or end-users by publishers or media producers ** Content industry, an umbrella term that encompasses companies owning and providing mas ...
. 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 In mathematical analysis, a Banach limit is a continuous linear functional \phi: \ell^\infty \to \mathbb defined on the Banach space \ell^\infty of all bounded complex-valued sequences such that for all sequences x = (x_n), y = (y_n) in \ell^\in ...
s, the dual of L^\infty and the
Stone–Čech compactification In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Ston ...
. All these are linked in one way or another to the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. Contents remain useful in certain technical problems in
geometric measure theory In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
; this is the theory of
Banach measure In the mathematics, mathematical discipline of measure theory, a Banach measure is a certain type of finite measure, content used to formalize geometric area in problems vulnerable to the axiom of choice. Traditionally, intuitive notions of are ...
s. A
charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...
is a generalization in both directions: it is a finitely additive, signed measure. (Cf.
ba space In mathematics, the ba space ba(\Sigma) of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive signed measures on \Sigma. The norm is defined as the variation, that is \, \nu\, =, \nu, (X). If Σ is ...
for information about ''bounded'' charges, where we say a charge is ''bounded'' to mean its range its a bounded subset of ''R''.)


See also

*
Abelian von Neumann algebra In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute. The prototypical example of an abelian von Neumann algebra is the algebra ''L''∞(''X'', μ) for μ a ...
*
Almost everywhere In measure theory (a branch of mathematical analysis), 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 ...
*
Carathéodory's extension theorem In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets ''R'' of a given set ''Ω'' can be extended to a measure on the σ- ...
*
Content (measure theory) 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 \mu defined on a collection of subsets \mathcal such that # ...
*
Fubini's theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
*
Fatou's lemma In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou. Fatou's lemm ...
*
Fuzzy measure theory In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (also ''capacity'', see ), whic ...
*
Geometric measure theory In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
*
Hausdorff measure In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that as ...
*
Inner measure In mathematics, in particular in measure theory, an inner measure is a function on the power set of a given set, with values in the extended real numbers, satisfying some technical conditions. Intuitively, the inner measure of a set is a lower bou ...
*
Lebesgue integration In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
* Lebesgue measure *
Lorentz space In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s,G. Lorentz, "On the theory of spaces Λ", ''Pacific Journal of Mathematics'' 1 (1951), pp. 411-429. are generalisations of the more familiar L^ spaces. The Lor ...
*
Lifting theory 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 Tu ...
*
Measurable cardinal In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivisi ...
* Measurable function *
Minkowski content The Minkowski content (named after Hermann Minkowski), or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a smooth curve in the plane, and area of a smooth ...
*
Outer measure In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer mea ...
*
Product measure In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology of tw ...
*
Pushforward measure In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function. Definition Given meas ...
*
Regular measure 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. Definition Let (''X'', ''T'') be a topolo ...
*
Vector measure In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only. Definitions and ...
*
Valuation (measure theory) In measure theory, or at least in the approach to it via the domain theory, a valuation is a Map (mathematics), map from the class of open sets of a topological space to the set of positive number, positive real numbers including infinity, with cert ...
*
Volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...


Notes


Bibliography

*
Robert G. Bartle Robert Gardner Bartle (November 20, 1927 – September 18, 2003) was an American mathematician specializing in real analysis. He is known for writing the popular textbooks ''The Elements of Real Analysis'' (1964), ''The Elements of Integration'' ...
(1995) ''The Elements of Integration and Lebesgue Measure'', Wiley Interscience. * * * * * Chapter III. * R. M. Dudley, 2002. ''Real Analysis and Probability''. Cambridge University Press. * * * Federer, Herbert. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969 xiv+676 pp. * Second printing. * * * R. Duncan Luce and Louis Narens (1987). "measurement, theory of," ''The New Palgrave: A Dictionary of Economics'', v. 3, pp. 428–32. * * ** The first edition was published with ''Part B: Functional Analysis'' as a single volume: * M. E. Munroe, 1953. ''Introduction to Measure and Integration''. Addison Wesley. * * * First printing. Note that there is a later (2017) second printing. Though usually there is little difference between the first and subsequent printings, in this case the second printing not only deletes from page 53 the Exercises 36, 40, 41, and 42 of Chapter 2 but also offers a (slightly, but still substantially) different presentation of part (ii) of Exercise 17.8. (The second printing's presentation of part (ii) of Exercise 17.8 (on the Luther decomposition) agrees with usual presentations, whereas the first printing's presentation provides a fresh perspective.) * Shilov, G. E., and Gurevich, B. L., 1978. ''Integral, Measure, and Derivative: A Unified Approach'', Richard A. Silverman, trans. Dover Publications. . Emphasizes the
Daniell integral In mathematics, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the Riemann integral to which students are typically first introduced. One of the main difficulties with the traditional f ...
. * * *


References


External links

*
Tutorial: Measure Theory for Dummies
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