# Measure (mathematics)

measuremeasure theorymeasurablemeasuresmeasurable setmeasure-theoreticmeasure spacemeasurable subsetmeasure theoreticpositive measure
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.wikipedia
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It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets is equal to the sum of the measures of the "smaller" subsets.
In mathematics, additivity (specifically finite additivity) and sigma additivity (also called countable additivity) of a function (often a measure) defined on subsets of a given set are abstractions of how intuitive properties of size (length, area, volume) of a set sum when considering multiple object, e.g. two apples are twice as much fruit as one apple.

### Émile Borel

BorelBorel EBorel, Félix Édouard Justin Émile
Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others.
As a mathematician, he was known for his founding work in the areas of measure theory and probability.

### Counting measure

In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure.
In mathematics, 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 ∞ if the subset is infinite.

### Probability theory

theory of probabilityprobabilityprobability theorist
The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures.
Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space.

### Signed measure

, then is called a signed measure.
In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values.

### Henri Lebesgue

Lebesguedoctoral dissertationHenri Léon Lebesgue
Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others.
At the same time he started his graduate studies at the Sorbonne, where he learned about Émile Borel's work on the incipient measure theory and Camille Jordan's work on the Jordan measure.

### Extended real number line

extended real line+∞extended real numbers
A function from to the extended real number line is called a measure if it satisfies the following properties:
It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration.

### Mathematical analysis

analysisclassical analysisanalytic
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures.
and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

### Integral

integrationintegral calculusdefinite integral
In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral.
Although all bounded piecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of Fourier analysis—to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory (a subfield of real analysis).

### Measurable space

measurable spacesmeasurablemeasurable state space
is called a measurable space, the members of Σ are called measurable sets. If and are two measurable spaces, then a function f : X \to Y is called measurable if for every
In mathematics, a measurable space or Borel space is a basic object in measure theory.

Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure, Young measure, and Loeb 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 sets, and inner regular on open sets.

### Measure space

measureprobability spacemeasure spaces.
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.

### Probability measure

measureprobability distributionlaw
A probability measure is a measure with total measure one – i.e.
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.

### Measurable function

measurableLebesgue measurableΣ-measurable
In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open.

### Borel measure

BorelBorel measurablemeasure
Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure, Young measure, and Loeb 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).

### Complete measure

completecomplete measure spacecompletion
is a complete translation-invariant measure on a σ-algebra containing the intervals in
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).

### Mass

inertial massgravitational massweight
In physics an example of a measure is spatial distribution of mass (see e.g., gravity potential), or another non-negative extensive property, conserved (see conservation law for a list of these) or not.
Mass is both a property of a physical body and a measure of its resistance to acceleration (a change in its state of motion) when a net force is applied.

### Baire measure

Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure, Young measure, and Loeb 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.

### Dynamical system

dynamical systemsdynamic systemdynamic systems
Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.
Combining insights from physics on the ergodic hypothesis with measure theory, this theorem solved, at least in principle, a fundamental problem of statistical mechanics.

### Euler measure

Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure, Young measure, and Loeb measure.
In measure theory, the Euler measure of a polyhedral set equals the Euler integral of its indicator function.

### Angle

acute angleacuteoblique
Circular angle measure is invariant under rotation, and hyperbolic angle measure is invariant under squeeze mapping.
Angle is also used to designate the measure of an angle or of a rotation.

### Young measure

Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure, Young measure, and Loeb 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.

### Haar measure

unimodularHaar measuresaverage over all possible rotations
The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties.
This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral".