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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 Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
,
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 op ...
,
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). ...
) 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 ele ...
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 speaking, ...
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. Far-reaching generalizations (such as spectral measures and projection-valued measures) 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 civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cu ...
, 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, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others.


Definition

Let X be a set and \Sigma a \sigma-algebra over X. A set function \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 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''. The pair (X, \Sigma) is called a '' measurable space'', and the members of \Sigma are called measurable sets. A triple (X, \Sigma, \mu) is called a ''
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
''. A probability measure is a measure with total measure one – that is, \mu(X) = 1. A probability space 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 poin ...
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 measures. 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 function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s 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.


Instances

Some important measures are listed here. * The counting measure is defined by \mu(S) = number of elements in S. * The
Lebesgue measure 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 wi ...
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 ...
measure is invariant under rotation, and hyperbolic angle measure is invariant under squeeze mapping. * 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 ...
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 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 probabil ...
. * 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,
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, Radon measure,
Young measure In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. They are a quantification of the oscillation effect of the sequence in the limi ...
, and Loeb measure. 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 ele ...
(see for example, gravity potential), or another non-negative extensive property, conserved (see
conservation law In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
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 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 hea ...
.


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
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 called ...
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 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 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 In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathema ...
t \in X. This property is used in connection with Lebesgue integral.


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 considered ...
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 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 wi ...
are σ-finite but not finite. Consider the closed intervals , 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, 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 ...
^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 In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
.


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 Euclidea ...
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 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 Formally ...
''. Observe, however, that complex measure is necessarily of finite variation, hence complex measures include finite signed measures but not, for example, the
Lebesgue measure 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 wi ...
. 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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
is called a '' projection-valued measure''; 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 defi ...
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 be ...
. 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. 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; this is the theory of
Banach measure In the mathematical discipline of measure theory, a Banach measure is a certain type of content used to formalize geometric area in problems vulnerable to the axiom of choice. Traditionally, intuitive notions of area are formalized as a class ...
s. A charge 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 * Almost everywhere * Carathéodory's extension theorem * Content (measure theory) * Fubini's theorem *
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 le ...
* Fuzzy measure theory * Geometric measure theory *
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 ...
* Inner measure *
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 Le ...
*
Lebesgue measure 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 wi ...
* Lorentz space * Lifting theory * Measurable cardinal * 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 smoot ...
* Outer measure * Product measure * Pushforward measure * Regular measure * Vector measure *
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


Notes


Bibliography

* Robert G. Bartle (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. * * *


References


External links

*
Tutorial: Measure Theory for Dummies
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