Measure theory

pseudo-function δ such that an integral of δ(x-c)f(x) always takes the value of f(c)

function assigning numbers to some subsets of a set, which could be seen as a generalization of length, area, volume and integral

real number strictly greater than zero

In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with area or volume. In probability theory, they are used to define events for which a probability can be defined. In this way, σ-algebras help to formalize the notion of size.

fractal and set of points on a line segment

function that returns 1 if an element is present in a specified subset and 0 if absent; naturally isomorphic with a set's subsets

theorem that there exists a decomposition of a unit solid ball into a finite number of disjoint subsets, which can be put back together in a different way to yield two identical copies of the unit sphere

Method of integration

function that is continuous everywhere but differentiable nowhere

triangle inequality of Lp space

function spaces generalizing finite-dimensional p norm spaces

function whose range is a subset of the real numbers

function between measurable spaces

ordered pair associating a set with a sigma-algebra, on which it is then possible to define a measure

continuous function that is not absolutely continuous

construct related to weighted sums and averages

subset of [0,1] whose intersection with any translation of the set of rationals is a singleton; elementary example of a non-Lebesgue-measurable set of reals

form of continuity for functions

subset of a measure space that is contained in a measurable set of measure zero

notion of convergence in mathematics

measure of similarity and diversity between sets

the mathematical study of optimal transportation and allocation of resources

set on which a generalization of volumes and integrals is defined; measurable space with a fixed measure

term used to describe a property on a set that is false only on a measurable set with zero measure

complex function on a measurable space that is piecewise constant with a finite number of measurable regions

set that is nowhere dense (in particular it contains no intervals), yet has positive measure

Set closed under union and intersection

mapping function

real function with finite total variation

function which is integrable on every compact subset of its domain of definition

set which cannot be assigned a meaningful "size"

concepts in probability mathematics

a statistic used for comparing the similarity of two samples

indexed set of subobjects of an algebraic structure

In mathematics, a nonempty collection of sets is called a -ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.

a measurable set with positive measure that contains no subset of smaller positive measure

distance function defined between probability distributions

paradox in mathematics

method for integrating a function with respect to its volume in different coordinate system

infimum and supremum almost everywhere

given a Borel measure, the set of those points whose neighbourhoods always have positive measure

certain metric on space of finite measures

study of geometric properties of sets through measure theory, extending differential geometry to not necessarily smooth sets

In mathematics, a -system (or pi-system) on a set \Omega is a collection P of certain subsets of \Omega, such that

theory of irregularities of distribution

mathematics concept

extended-nonnegative-real-valued function defined on a field of sets (or more generally, a boolean algebra) that is additive over finitely many disjoint sets

subject of study in ergodic theory

separable Banach space equipped with a Hilbert subspace such that the standard cylinder set measure on the Hilbert subspace induces a Gaussian measure on the whole Banach space

Result in integral geometry

set theory construction

concept in measure theory

Type of mathematical function

property in the mathematical theory of stochastic processes

In mathematics, a non-empty collection of sets \mathcal{R} is called a -ring (pronounced "") if it is closed under union, relative complementation, and countable intersection. The name "delta-ring" originates from the German word for intersection, "Durchschnitt", which is meant to highlight the ring's closure under countable intersection, in contrast to a -ring which is closed under countable unions.

heuristics in measure theory

function whose domain is a collection of sets