In
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 Interna ...
,
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
A shape or figure is a graphics, graphical representation of an obje ...
,
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). The de ...
) 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 elementar ...
and
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), 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 ...
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 o ...
,
integration theory, and can be generalized to assume
negative values, as with
electrical charge. Far-reaching generalizations (such as
spectral measure
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his disse ...
s and
projection-valued measures) of measure are widely used in
quantum physics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
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 cult ...
, when
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
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
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Math ...
,
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
Nikolai Nikolaevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlaɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 January 1950) was a Soviet/Ru ...
,
Johann Radon,
Constantin Carathéodory, and
Maurice Fréchet Maurice may refer to:
People
*Saint Maurice (died 287), Roman legionary and Christian martyr
*Maurice (emperor) or Flavius Mauricius Tiberius Augustus (539–602), Byzantine emperor
*Maurice (bishop of London) (died 1107), Lord Chancellor and Lo ...
, among others.
Definition
Let
be a set and
a
-algebra over
A
set function from
to the
extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra ...
is called a measure if it satisfies the following properties:
*Non-negativity: For all
in
we have
*Null empty set:
*Countable additivity (or
-additivity): For all
countable collections
of pairwise
disjoint sets
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A ...
in Σ,
If at least one set
has finite measure, then the requirement that
is met automatically. Indeed, by countable additivity,
and therefore
If the condition of non-negativity is omitted but the second and third of these conditions are met, and
takes on at most one of the values
then
is called a ''
signed measure''.
The pair
is called a ''
measurable space'', and the members of
are called measurable sets.
A
triple
Triple is used in several contexts to mean "threefold" or a "treble":
Sports
* Triple (baseball), a three-base hit
* A basketball three-point field goal
* A figure skating jump with three rotations
* In bowling terms, three strikes in a row
* In ...
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 i ...
''. 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 a probability measure and the more gener ...
is a measure with total measure one – that is,
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 points ...
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 (38 ...
(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 o ...
) 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 vec ...
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 value ...
s with
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
. 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
= number of elements in
* 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 wit ...
on
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
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by .
In physics and mathematics, continuous translational symmetry is the invariance of a system of equatio ...
measure on a ''σ''-algebra containing the
intervals in
such that
; and every other measure with these properties extends Lebesgue measure.
* Circular
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
measure is invariant under
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
, and
hyperbolic angle measure is invariant under
squeeze mapping.
* The
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 was introduced by Alfréd Haar in 1933, though ...
for a
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
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 str ...
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 ass ...
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
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
. 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 probabili ...
.
* The
Dirac measure ''δ''
''a'' (cf.
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
) is given by ''δ''
''a''(''S'') = ''χ''
''S''(a), where ''χ''
''S'' is the
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 is a subset of some set , one has \mathbf_(x)=1 if x\i ...
of
The measure of a set is 1 if it contains the point
and 0 otherwise.
Other 'named' measures used in various theories include:
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 restrictions on the measure, as described below.
F ...
,
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, or parallelepiped.
It turns out that for ...
,
ergodic measure
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies t ...
,
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 named ...
,
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 Baire sets are the same, so Baire measures are the ...
,
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 mathematics, a Loeb space is a type of measure space introduced by using nonstandard analysis.
Construction
Loeb's construction starts with a finitely additive map \nu from an internal algebra \mathcal A of sets to the nonstandard reals. Def ...
.
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 elementar ...
(see for example,
gravity potential), or another non-negative
extensive property
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...
,
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 heat b ...
.
Basic properties
Let
be a measure.
Monotonicity
If
and
are measurable sets with
then
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 calle ...
of (not necessarily disjoint) measurable sets
in
Continuity from below
If
are measurable sets that are increasing (meaning that
) then the
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of the sets
is measurable and
Continuity from above
If
are measurable sets that are decreasing (meaning that
) then the
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
of the sets
is measurable; furthermore, if at least one of the
has finite measure then
This property is false without the assumption that at least one of the
has finite measure. For instance, for each
let
which all have infinite Lebesgue measure, but the intersection is empty.
Other properties
Completeness
A measurable set
is called a ''null set'' if
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
which differ by a negligible set from a measurable set
that is, such that the
symmetric difference of
and
is contained in a null set. One defines
to equal
μ = μ (a.e.)
If