Individual pieces set
In the theory of fair cake-cutting, the individual-pieces set (IPS)is a geometric object that represents all possible utility vectors in cake partitions.
Borel isomorphism
In mathematics, a Borel isomorphism is a measurable bijective function between two measurable standard Borel spaces. By Souslin's theorem in standard Borel spaces (a set that is both analytic and coan
Laplacian of the indicator
In mathematics, the Laplacian of the indicator of the domain D is a generalisation of the derivative of the Dirac delta function to higher dimensions, and is non-zero only on the surface of D. It can
Indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if A is a subset
Solovay model
In the mathematical field of set theory, the Solovay model is a model constructed by Robert M. Solovay in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of ch
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 o
Ruziewicz problem
In mathematics, the Ruziewicz problem (sometimes Banach–Ruziewicz problem) in measure theory asks whether the usual Lebesgue measure on the n-sphere is characterised, up to proportionality, by its pro
Intensity (measure theory)
In the mathematical discipline of measure theory, the intensity of a measure is the average value the measure assigns to an interval of length one.
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
Volterra's function
In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function V defined on the real line R with the following curious combination of properties:
* V is differentiable everyw
Finite-dimensional distribution
In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure (or process)
Filtration (mathematics)
In mathematics, a filtration is an indexed family of subobjects of a given algebraic structure , with the index running over some totally ordered index set , subject to the condition that if in , then
Nikodym set
In mathematics, a Nikodym set is a subset of the unit square in with complement of Lebesgue measure zero, such that, given any point in the set, there is a straight line that only intersects the set a
Infinite-dimensional Lebesgue measure
In mathematics, there is a theorem stating that there is no analogue of Lebesgue measure on an infinite-dimensional Banach space. Other kinds of measures are therefore used on infinite-dimensional spa
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.
Jaccard index
The Jaccard index, also known as the Jaccard similarity coefficient, is a statistic used for gauging the similarity and diversity of sample sets. It was developed by Grove Karl Gilbert in 1884 as his
Sigma-ideal
In mathematics, particularly measure theory, a 𝜎-ideal, or sigma ideal, of a sigma-algebra (𝜎, read "sigma," means countable in this context) is a subset with certain desirable closure properties. It
Null set
In mathematical analysis, a null set is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total leng
Regular conditional probability
In probability theory, regular conditional probability is a concept that formalizes the notion of conditioning on the outcome of a random variable. The resulting conditional probability distribution i
Transportation theory (mathematics)
In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mat
Convergence in measure
Convergence in measure is either of two distinct mathematical concepts both of which generalizethe concept of convergence in probability.
Concentration of measure
In mathematics, concentration of measure (about a median) is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space th
Baire set
In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets. There are several inequivalen
Σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersecti
Absolute continuity
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations o
Rectifiable set
In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifia
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henr
Talagrand's concentration inequality
In the probability theory field of mathematics , Talagrand's concentration inequality is an isoperimetric-type inequality for product probability spaces. It was first proved by the French mathematicia
Tightness of measures
In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity".
Radonifying function
In measure theory, a radonifying function (ultimately named after Johann Radon) between measurable spaces is one that takes a cylinder set measure (CSM) on the first space to a true measure on the sec
Caccioppoli set
In mathematics, a Caccioppoli set is a set whose boundary is measurable and has (at least locally) a finite measure. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli s
Clarkson's inequalities
In mathematics, Clarkson's inequalities, named after James A. Clarkson, are results in the theory of Lp spaces. They give bounds for the Lp-norms of the sum and difference of two measurable functions
Measurable group
In mathematics, a measurable group is a special type of group in the intersection between group theory and measure theory. Measurable groups are used to study measures is an abstract setting and are o
Bounded variation
In mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property
Ring of sets
In mathematics, there are two different notions of a ring of sets, both referring to certain families of sets. In order theory, a nonempty family of sets is called a ring (of sets) if it is closed und
Convergence of measures
In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence of measures, consider a seque
Standard probability space
In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assu
Curvature of a measure
In mathematics, the curvature of a measure defined on the Euclidean plane R2 is a quantification of how much the measure's "distribution of mass" is "curved". It is related to notions of curvature in
Differentiation of integrals
In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approxima
Luzin N property
In mathematics, a function f on the interval [a, b] has the Luzin N property, named after Nikolai Luzin (also called Luzin property or N property) if for all such that , there holds: , where stands fo
Pi-system
In mathematics, a π-system (or pi-system) on a set is a collection of certain subsets of such that
* is non-empty.
* If then That is, is a non-empty family of subsets of that is closed under non-emp
Sierpiński set
In mathematics, a Sierpiński set is an uncountable subset of a real vector space whose intersection with every measure-zero set is countable. The existence of Sierpiński sets is independent of the axi
Ba space
In mathematics, the ba space of an algebra of sets is the Banach space consisting of all bounded and finitely additive signed measures on . The norm is defined as the variation, that is If Σ is a sigm
Projection (measure theory)
In measure theory, projection maps often appear when working with product spaces: The product sigma-algebra of measurable spaces is defined to be the finest such that the projection mappings will be m
Euler calculus
Euler calculus is a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently by integrating with respect to the Euler characteristic
Universally measurable set
In mathematics, a subset of a Polish space is universally measurable if it is measurable with respect to every complete probability measure on that measures all Borel subsets of . In particular, a uni
Falconer's conjecture
In geometric measure theory, Falconer's conjecture, named after Kenneth Falconer, is an unsolved problem concerning the sets of Euclidean distances between points in compact -dimensional spaces. Intui
Integration by parts operator
In mathematics, an integration by parts operator is a linear operator used to formulate integration by parts formulae; the most interesting examples of integration by parts operators occur in infinite
Aubin–Lions lemma
In mathematics, the Aubin–Lions lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a compactness criterion that is useful in the study of
Klee's measure problem
In computational geometry, Klee's measure problem is the problem of determining how efficiently the measure of a union of (multidimensional) rectangular ranges can be computed. Here, a d-dimensional r
Simple function
In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice
Sigma-ring
In mathematics, a nonempty collection of sets is called a 𝜎-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.
Measure algebra
In mathematics, a measure algebra is a Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure algebra on the Boolean algebra of measurable
Distribution function (measure theory)
In mathematics, a distribution function is a real function in measure theory. From every measure on the algebra of Borel sets of real numbers, a distribution function can be constructed, which reflect
Effective dimension
In mathematics, effective dimension is a modification of Hausdorff dimension and other fractal dimensions that places it in a computability theory setting. There are several variations (various notion
Carathéodory's criterion
Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory that characterizes when a set is Lebesgue measurable.
Coarea formula
In the mathematical field of geometric measure theory, the coarea formula expresses the integral of a function over an open set in Euclidean space in terms of integrals over the level sets of another
Euler measure
In measure theory, the Euler measure of a polyhedral set equals the Euler integral of its indicator function.
Progressively measurable process
In mathematics, progressive measurability is a property in the theory of stochastic processes. A progressively measurable process, while defined quite technically, is important because it implies the
Planar lamina
In mathematics, a planar lamina (or plane lamina) is a figure representing a thin, usually uniform, flat layer of the solid. It serves also as an idealized model of a planar cross section of a solid b
Wasserstein metric
In mathematics, the Wasserstein distance or Kantorovich–Rubinstein metric is a distance function defined between probability distributions on a given metric space . It is named after Leonid Vaseršteĭn
Crofton formula
In mathematics, the Crofton formula, named after Morgan Crofton (1826–1915), is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line int
Abstract Wiener space
The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite-dimensional spaces. The construction emph
Homological integration
In the mathematical fields of differential geometry and geometric measure theory, homological integration or geometric integration is a method for extending the notion of the integral to manifolds. Ra
Weierstrass function
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its di
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 th
Hausdorff density
In measure theory, a field of mathematics, the Hausdorff density measures how concentrated a Radon measure is at some point.
Strong measure zero set
In mathematical analysis, a strong measure zero set is a subset A of the real line with the following property: for every sequence (εn) of positive reals there exists a sequence (In) of intervals such
Von Neumann paradox
In mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar figure such as the unit square into sets of points and subject each set to an area-preser
Sigma-additive set function
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, If
Hanner's inequalities
In mathematics, Hanner's inequalities are results in the theory of Lp spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of Lp spaces
Measurable acting group
In mathematics, a measurable acting group is a special group that acts on some space in a way that is compatible with structures of measure theory. Measurable acting groups are found in the intersecti
Positive real numbers
In mathematics, the set of positive real numbers, is the subset of those real numbers that are greater than zero. The non-negative real numbers, also include zero. Although the symbols and are ambiguo
Positive and negative sets
In measure theory, given a measurable space and a signed measure on it, a set is called a positive set for if every -measurable subset of has nonnegative measure; that is, for every that satisfies hol
Smith–Volterra–Cantor set
In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), y
Essential infimum and essential supremum
In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deal
Atom (measure theory)
In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller positive measure. A measure which has no atoms is called non-ato
Hamburger moment problem
In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence (m0, m1, m2, ...), does there exist a positive Borel measure μ (for instance
Cartan–Hadamard conjecture
In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized
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 x-axis. The Lebesgue integral
Hopf decomposition
In mathematics, the Hopf decomposition, named after Eberhard Hopf, gives a canonical decomposition of a measure space (X, μ) with respect to an invertible non-singular transformation T:X→X, i.e. a tra
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 numb
Christ–Kiselev maximal inequality
In mathematics, the Christ–Kiselev maximal inequality is a maximal inequality for filtrations, named for mathematicians Michael Christ and Alexander Kiselev.
Computable measure theory
In mathematics, computable measure theory is the part of computable analysis that deals with effective versions of measure theory.
Sugeno integral
In mathematics, the Sugeno integral, named after M. Sugeno, is a type of integral with respect to a fuzzy measure. Let be a measurable space and let be an -measurable function. The Sugeno integral ove
Prevalent and shy sets
In mathematics, the notions of prevalence and shyness are notions of "almost everywhere" and "measure zero" that are well-suited to the study of infinite-dimensional spaces and make use of the transla
Essential range
In mathematics, particularly measure theory, the essential range, or the set of essential values, of a function is intuitively the 'non-negligible' range of the function: It does not change between tw
Sørensen–Dice coefficient
The Sørensen–Dice coefficient (see below for other names) is a statistic used to gauge the similarity of two samples. It was independently developed by the botanists Thorvald Sørensen and Lee Raymond
Measurable function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of an
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
Simple matching coefficient
The simple matching coefficient (SMC) or Rand similarity coefficient is a statistic used for comparing the similarity and diversity of sample sets. Given two objects, A and B, each with n binary attri
Bochner measurable function
In mathematics – specifically, in functional analysis – a Bochner-measurable function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable
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 m
Distortion (mathematics)
In mathematics, the distortion is a measure of the amount by which a function from the Euclidean plane to itself distorts circles to ellipses. If the distortion of a function is equal to one, then it
Lévy–Prokhorov metric
In mathematics, the Lévy–Prokhorov metric (sometimes known just as the Prokhorov metric) is a metric (i.e., a definition of distance) on the collection of probability measures on a given metric space.
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compa
Tversky index
The Tversky index, named after Amos Tversky, is an asymmetric similarity measure on sets that compares a variant to a prototype. The Tversky index can be seen as a generalization of the Sørensen–Dice
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 defined on
Approximate tangent space
In geometric measure theory an approximate tangent space is a measure theoretic generalization of the concept of a tangent space for a differentiable manifold.
Malliavin's absolute continuity lemma
In mathematics — specifically, in measure theory — Malliavin's absolute continuity lemma is a result due to the French mathematician Paul Malliavin that plays a foundational rôle in the regularity (sm
Direct integral
In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of v
H-derivative
In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus.
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
Non-measurable set
In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, a
Information theory and measure theory
This article discusses how information theory (a branch of mathematics studying the transmission, processing and storage of information) is related to measure theory (a branch of mathematics related t
Conull set
In measure theory, a conull set is a set whose complement is null, i.e., the measure of the complement is zero. For example, the set of irrational numbers is a conull subset of the real line with Lebe
Varifold
In mathematics, a varifold is, loosely speaking, a measure-theoretic generalization of the concept of a differentiable manifold, by replacing differentiability requirements with those provided by rect
Weight function
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The resu
Overlap coefficient
The overlap coefficient, or Szymkiewicz–Simpson coefficient, is a similarity measure that measures the overlap between two finite sets. It is related to the Jaccard index and is defined as the size of
Minkowski inequality
In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let S be a measure space, let 1 ≤ p < ∞ and let f and g be elements of Lp(S). Then f + g is
Delta-ring
In mathematics, a non-empty collection of sets is called a δ-ring (pronounced "delta-ring") if it is closed under union, relative complementation, and countable intersection. The name "delta-ring" ori
Weakly measurable function
In mathematics—specifically, in functional analysis—a weakly measurable function taking values in a Banach space is a function whose composition with any element of the dual space is a measurable func
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 Doro
Equivalence (measure theory)
In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero.
Discrepancy theory
In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the
Measure-preserving dynamical system
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poin
Standard Borel space
In mathematics, a standard Borel space is the Borel space associated to a Polish space. Discounting Borel spaces of discrete Polish spaces, there is, up to isomorphism of measurable spaces, only one s
Vague topology
In mathematics, particularly in the area of functional analysis and topological vector spaces, the vague topology is an example of the weak-* topology which arises in the study of measures on locally
Loeb space
In mathematics, a Loeb space is a type of measure space introduced by Loeb using nonstandard analysis.
Volume element
In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volu
Lévy metric
In mathematics, the Lévy metric is a metric on the space of cumulative distribution functions of one-dimensional random variables. It is a special case of the Lévy–Prokhorov metric, and is named after
Continuity set
In measure theory, a branch of mathematics, a continuity set of a measure μ is any Borel set B such that where is the (topological) boundary of B. For signed measures, one asks that The class of all c
Dirac delta function
In mathematics, the Dirac delta distribution (δ distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at
Cylindrical σ-algebra
In mathematics — specifically, in measure theory and functional analysis — the cylindrical σ-algebra or product σ-algebra is a type of σ-algebra which is often used when studying product measures or p
Brezis–Lieb lemma
In the mathematical field of analysis, the Brezis–Lieb lemma is a basic result in measure theory. It is named for Haïm Brézis and Elliott Lieb, who discovered it in 1983. The lemma can be viewed as an
Valuation (measure theory)
In measure theory, or at least in the approach to it via the domain theory, a valuation is a map from the class of open sets of a topological space to the set of positive real numbers including infini
Set-theoretic limit
In mathematics, the limit of a sequence of sets (subsets of a common set ) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the
Τ-additivity
In mathematics, in the field of measure theory, τ-additivity is a certain property of measures on topological spaces. A measure or set function on a space whose domain is a sigma-algebra is said to be
Cantor function
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions
Minkowski–Steiner formula
In mathematics, the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space. More precisely, it defines the surface area as the "derivative" o
Almgren–Pitts min-max theory
In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces. The theory started with the effo
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 differe
Locally integrable function
In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of d
Support (measure theory)
In mathematics, the support (sometimes topological support or spectrum) of a measure μ on a measurable topological space (X, Borel(X)) is a precise notion of where in the space X the measure "lives".
Vitali covering lemma
In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in t
Symmetric decreasing rearrangement
In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function.
Real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These se
Radon–Nikodym set
In the theory of fair cake-cutting, the Radon–Nikodym set (RNS) is a geometric object that represents a cake, based on how different people evaluate the different parts of the cake.
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by Ge
Trigonometric moment problem
In mathematics, the trigonometric moment problem is formulated as follows: given a finite sequence {α0, ... αn }, does there exist a positive Borel measure μ on the interval [0, 2π] such that In other
Holmes–Thompson volume
In geometry of normed spaces, the Holmes–Thompson volume is a notion of volume that allows to compare sets contained in different normed spaces (of the same dimension). It was introduced by Raymond D.