In
mathematics, the concept of a measure is a generalization and formalization of
geometrical measures (
length,
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 or planar lamina, while ''surface area'' refers to the area of an open s ...
,
volume) and other common notions, such as
mass and
probability 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 ...
,
integration theory, and can be generalized to assume
negative values, as with
electrical charge
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
. 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 dis ...
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, q ...
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 i ...
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,
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,
Johann Radon,
Constantin Carathéodory
Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
, and
Maurice Fréchet, among others.
Definition
Let
be a set and
a
-algebra over
A
set function from
to the
extended real number line 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
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
collections
of pairwise
disjoint sets 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
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.
Definition
Consider a set X and a σ-algebra \mathcal A on X. Then t ...
'', and the members of
are called measurable sets.
A
triple is called a ''
measure space''. 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 ge ...
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 po ...
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 (3 ...
(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 ...
) are
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 sets, outer regular on all Borel s ...
s. Radon measures have an alternative definition in terms of linear functionals on the
locally convex topological vector space of
continuous functions 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 smalle ...
. This approach is taken by
Bourbaki (2004) and a number of other sources. For more details, see the article on
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 sets, outer regular on all Borel s ...
s.
Instances
Some important measures are listed here.
* The
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 subset has finitely many elements, and infini ...
is defined by
= number of elements in
* The
Lebesgue measure 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 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 rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
measure is invariant under
rotation, and
hyperbolic angle measure is invariant under
squeeze mapping
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping.
For a fixed positive real number , th ...
.
* The
Haar measure 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 ...
topological group 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 ...
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 , 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 o ...
. Such a measure is called a ''probability measure''. See
probability axioms.
* 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 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. ...
,
Jordan measure,
ergodic measure,
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 nam ...
,
Baire measure,
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 sets, outer regular on all Borel s ...
,
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 physics an example of a measure is spatial distribution of
mass (see for example,
gravity potential
In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move an object to that location from a fixed reference location. It is analogous to the electric p ...
), or another non-negative
extensive property,
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
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
, known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics.
*
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 of the canonical ensemble to infinite systems.
Th ...
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
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
sequence 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
** ''U ...
of the sets
is measurable and
Continuity from above
If
are measurable sets that are decreasing (meaning that
) then the
intersection 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