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 which
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 additiona
Tangent measure
In measure theory, tangent measures are used to study the local behavior of Radon measures, in much the same way as tangent spaces are used to study the local behavior of differentiable manifolds. Tan
Invariant measure
In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbol
Complete measure
In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measu
Uniformly distributed measure
In mathematics — specifically, in geometric measure theory — a uniformly distributed measure on a metric space is one for which the measure of an open ball depends only on its radius and not on its ce
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 spac
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
Discrete measure
In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if it is concentrated on an at most countable set. The su
Besov measure
In mathematics — specifically, in the fields of probability theory and inverse problems — Besov measures and associated Besov-distributed random variables are generalisations of the notions of Gaussia
Trivial measure
In mathematics, specifically in measure theory, the trivial measure on any measurable space (X, Σ) is the measure μ which assigns zero measure to every measurable set: μ(A) = 0 for all A in Σ.
Inner regular measure
In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.
Two-dimensional Yang–Mills theory
In mathematical physics, two-dimensional Yang–Mills theory is the special case of Yang–Mills theory in which the dimension of spacetime is taken to be two. This special case allows for a rigorously de
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
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 out
Quasi-invariant measure
In mathematics, a quasi-invariant measure μ with respect to a transformation T, from a measure space X to itself, is a measure which, roughly speaking, is multiplied by a numerical function of T. An i
Saturated measure
In mathematics, a measure is said to be saturated if every locally measurable set is also measurable. A set , not necessarily measurable, is said to be a locally measurable set if for every measurable
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
Σ-finite measure
In mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞), and a set A in Σ is of finit
Resource bounded measure
Lutz's resource bounded measure is a generalisation of Lebesgue measure to complexity classes. It was originally developed by Jack Lutz. Just as Lebesgue measure gives a method to quantify the size of
Strictly positive measure
In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that is zero "only on points".
G-measure
In mathematics, a G-measure is a measure that can be represented as the weak-∗ limit of a sequence of measurable functions . A classic example is the Riesz product where . The weak-∗ limit of this pro
Intensity measure
In probability theory, an intensity measure is a measure that is derived from a random measure. The intensity measure is a non-random measure and is defined as the expectation value of the random meas
Maximising measure
In mathematics — specifically, in ergodic theory — a maximising measure is a particular kind of probability measure. Informally, a probability measure μ is a maximising measure for some function f if
Perfect measure
In mathematics — specifically, in measure theory — a perfect measure (or, more accurately, a perfect measure space) is one that is "well-behaved" in some sense. Intuitively, a perfect measure μ is one
Random measure
In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such
Sub-probability measure
In the mathematical theory of probability and measure, a sub-probability measure is a measure that is closely related to probability measures. While probability measures always assign the value 1 to t
Isotropic measure
In probability theory, an isotropic measure is any mathematical measure that is invariant under linear isometries. It is a standard simplification and assumption used in probability theory. Generally,
Decomposable measure
In mathematics, a decomposable measure (also known as a strictly localizable measure) is a measure that is a disjoint union of finite measures. This is a generalization of σ-finite measures, which are
Empirical measure
In probability theory, an empirical measure is a random measure arising from a particular realization of a (usually finite) sequence of random variables. The precise definition is found below. Empiric
Green measure
In mathematics — specifically, in stochastic analysis — the Green measure is a measure associated to an Itō diffusion. There is an associated Green formula representing suitably smooth functions in te
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. Intuiti
Cylinder set measure
In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a measure on an infinite-dimensional vector space. An example is the Gaussian c
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,
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 to
Logarithmically concave measure
In mathematics, a Borel measure μ on n-dimensional Euclidean space is called logarithmically concave (or log-concave for short) if, for any compact subsets A and B of and 0 < λ < 1, one has where λ A
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 Bai
Idempotent measure
In mathematics, an idempotent measure on a is a probability measure that equals its convolution with itself; in other words, an idempotent measure is an idempotent element in the topological semigroup
Signed measure
In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values.
Borel regular measure
In mathematics, an outer measure μ on n-dimensional Euclidean space Rn is called a Borel regular measure if the following two conditions hold:
* Every Borel set B ⊆ Rn is μ-measurable in the sense of
Pre-measure
In mathematics, a pre-measure is a set function that is, in some sense, a precursor to a bona fide measure on a given space. Indeed, one of the fundamental theorems in measure theory states that a pre
Locally finite measure
In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.
Harmonic measure
In mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem. In probability
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, volum
Carleson measure
In mathematics, a Carleson measure is a type of measure on subsets of n-dimensional Euclidean space Rn. Roughly speaking, a Carleson measure on a domain Ω is a measure that does not vanish at the boun
Euler measure
In measure theory, the Euler measure of a polyhedral set equals the Euler integral of its indicator function.
Convex measure
In measure and probability theory in mathematics, a convex measure is a probability measure that — loosely put — does not assign more mass to any intermediate set "between" two measurable sets A and B
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.
Singular measure
In mathematics, two positive (or signed or complex) measures and defined on a measurable space are called singular if there exist two disjoint sets whose union is such that is zero on all measurable s
Finite measure
In measure theory, a branch of mathematics, a finite measure or totally finite measure is a special measure that always takes on finite values. Among finite measures are probability measures. The fini
Spherical measure
In mathematics — specifically, in geometric measure theory — spherical measure σn is the "natural" Borel measure on the n-sphere Sn. Spherical measure is often normalized so that it is a probability m
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
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. The difference between
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 su
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 o
Haar measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure
Rajchman measure
In mathematics, a Rajchman measure, studied by Rajchman, is a regular Borel measure on a locally compact group such as the circle, whose Fourier transform vanishes at infinity.
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
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".
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
S-finite measure
In measure theory, a branch of mathematics that studies generalized notions of volumes, an s-finite measure is a special type of measure. An s-finite measure is more general than a finite measure, but
Secondary measure
In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the ortho
Metric outer measure
In mathematics, a metric outer measure is an outer measure μ defined on the subsets of a given metric space (X, d) such that for every pair of positively separated subsets A and B of X.
Transverse measure
In mathematics, a measure on a real vector space is said to be transverse to a given set if it assigns measure zero to every translate of that set, while assigning finite and positive (i.e. non-zero)
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 usin
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".
Borel measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restr
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
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
Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimen