In mathematics, especially
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
, a set function is a
function whose
domain is a
family
Family (from ) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictabili ...
of
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s of some given set and that (usually) takes its values in the
extended real number line which consists of the
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s
and
A set function generally aims to subsets in some way.
Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.
Definitions
If
is a
family of sets
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection F of subsets of a given set S is called a family of su ...
over
(meaning that
where
denotes the
powerset) then a is a function
with
domain and
codomain vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
, as with
vector measures,
complex measures, and
projection-valued measures.
The domain of a set function may have any number properties; the commonly encountered properties and categories of families are listed in the table below.
In general, it is typically assumed that
is always
well-defined for all
or equivalently, that
does not take on both
and
as values. This article will henceforth assume this; although alternatively, all definitions below could instead be qualified by statements such as "whenever the sum/series is defined". This is sometimes done with subtraction, such as with the following result, which holds whenever
is
finitely additive:
::
is defined with
satisfying
and
Null sets
A set
is called a (with respect to
) or simply if
Whenever
is not identically equal to either
or
then it is typically also assumed that:
- : if
Variation and mass
The
is
where
denotes the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
(or more generally, it denotes the
norm or
seminorm
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
if
is vector-valued in a (
semi)
normed space).
Assuming that
then
is called the of
and
is called the of
A set function is called if for every
the value
is (which by definition means that
and
; an is one that is equal to
or
).
Every finite set function must have a finite
mass
Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
.
Common properties of set functions
A set function
on
is said to be
- if it is valued in
- if for all
pairwise disjoint
In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...
finite sequences such that
* If is closed under binary unions then is finitely additive if and only if for all disjoint pairs
* If is finitely additive and if then taking shows that which is only possible if or where in the latter case, for every (so only the case is useful).
- or if in addition to being finitely additive, for all
pairwise disjoint
In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...
sequences in such that all of the following hold:
-
* The series on the left hand side is defined in the usual way as the limit
* As a consequence, if is any
permutation
In mathematics, a permutation of a set can mean one of two different things:
* an arrangement of its members in a sequence or linear order, or
* the act or process of changing the linear order of an ordered set.
An example of the first mean ...
/bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
then this is because and applying this condition (a) twice guarantees that both and hold. By definition, a convergent series with this property is said to be unconditionally convergent. Stated in plain English, this means that rearranging/relabeling the sets to the new order does not affect the sum of their measures. This is desirable since just as the union does not depend on the order of these sets, the same should be true of the sums and
- if is not infinite then this series must also converge absolutely, which by definition means that must be finite. This is automatically true if is non-negative (or even just valued in the extended real numbers).
* As with any convergent series of real numbers, by the Riemann series theorem, the series converges absolutely if and only if its sum does not depend on the order of its terms (a property known as unconditional convergence). Since unconditional convergence is guaranteed by (a) above, this condition is automatically true if is valued in
- if is infinite then it is also required that the value of at least one of the series be finite (so that the sum of their values is well-defined). This is automatically true if is non-negative.
- a if it is non-negative, countably additive (including finitely additive), and has a null empty set.
- a if it is a pre-measure whose domain is a σ-algebra. That is to say, a measure is a non-negative countably additive set function on a σ-algebra that has a null empty set.
- a if it is a measure that has a
mass
Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
of
- an if it is non-negative, countably subadditive, has a null empty set, and has the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
as its domain.
* Outer measures appear in the Carathéodory's extension theorem and they are often restricted to Carathéodory measurable subsets
- a if it is countably additive, has a null empty set, and does not take on both and as values.
- if every subset of every null set is null; explicitly, this means: whenever and is any subset of then and
* Unlike many other properties, completeness places requirements on the set (and not just on 's values).
- if there exists a sequence in such that is finite for every index and also
- if there exists a subfamily of pairwise disjoint sets such that is finite for every and also (where ).
* Every -finite set function is decomposable although not conversely. For example, the counting measure on (whose domain is ) is decomposable but not -finite.
- a if it is a countably additive set function valued in a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(such as a normed space) whose domain is a σ-algebra.
* If is valued in a normed space then it is countably additive if and only if for any pairwise disjoint
In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...
sequence in If is finitely additive and valued in a Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
then it is countably additive if and only if for any pairwise disjoint sequence in
- a if it is a countably additive complex-valued set function whose domain is a σ-algebra.
* By definition, a complex measure never takes as a value and so has a null empty set.
- a if it is a measure-valued random element.
Arbitrary sums
As described
in this article's section on generalized series, for any family
of
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s indexed by an arbitrary
indexing set it is possible to define their sum
as the limit of the
net of finite partial sums
where the domain
is
directed
Direct may refer to:
Mathematics
* Directed set, in order theory
* Direct limit of (pre), sheaves
* Direct sum of modules, a construction in abstract algebra which combines several vector spaces
Computing
* Direct access (disambiguation), a ...
by
Whenever this
net converges then its limit is denoted by the symbols
while if this net instead diverges to
then this may be indicated by writing
Any sum over the empty set is defined to be zero; that is, if
then
by definition.
For example, if
for every
then
And it can be shown that
If
then the generalized series
converges in
if and only if
converges unconditionally (or equivalently,
converges absolutely) in the usual sense.
If a generalized series
converges in
then both
and
also converge to elements of
and the set
is necessarily
countable
In mathematics, a Set (mathematics), set is countable if either it is finite set, 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 fro ...
(that is, either finite or
countably infinite
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 numbe ...
);
this remains true if
is replaced with any
normed space.
[
It follows that in order for a generalized series to converge in or it is necessary that all but at most countably many will be equal to which means that is a sum of at most countably many non-zero terms.
Said differently, if is uncountable then the generalized series does not converge.
In summary, due to the nature of the real numbers and its topology, every generalized series of real numbers (indexed by an arbitrary set) that converges can be reduced to an ordinary absolutely convergent series of countably many real numbers. So in the context of measure theory, there is little benefit gained by considering uncountably many sets and generalized series. In particular, this is why the definition of " countably additive" is rarely extended from countably many sets in (and the usual countable series ) to arbitrarily many sets (and the generalized series ).
]
Inner measures, outer measures, and other properties
A set function is said to be/satisfies
- if whenever satisfy
- if it satisfies the following condition, known as : for all such that
* Every finitely additive function on a field of sets is modular.
* In geometry, a set function valued in some abelian semigroup that possess this property is known as a . This geometric definition of "valuation" should not be confused with the stronger non-equivalent measure theoretic definition of "valuation" that is given below.
- if for all such that
- if for all finite sequences that satisfy
- or if for all sequences in that satisfy
* If is closed under finite unions then this condition holds if and only if for all If is non-negative then the absolute values may be removed.
* If is a measure then this condition holds if and only if for all in If is a
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
then this inequality is Boole's inequality.
* If is countably subadditive and with then is finitely subadditive.
- if whenever are disjoint with
- if for all of sets in such that with and all finite.
* Lebesgue measure is continuous from above but it would not be if the assumption that all are eventually finite was omitted from the definition, as this example shows: For every integer let be the open interval so that where
- if for all of sets in such that
- if whenever satisfies then for every real there exists some such that and
- an if is non-negative, countably subadditive, has a null empty set, and has the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
as its domain.
- an if is non-negative, superadditive, continuous from above, has a null empty set, has the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
as its domain, and is approached from below.
- if every measurable set of positive measure contains an
atom
Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
.
If a binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, a binary operation ...
is defined, then a set function is said to be
- if for all and such that
Topology related definitions
If is a topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
on then a set function is said to be:
- a if it is a measure defined on the σ-algebra of all
Borel set
In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...
s, which is the smallest σ-algebra containing all open subsets (that is, containing ).
- a if it is a measure defined on the σ-algebra of all Baire sets.
- if for every point there exists some neighborhood of this point such that is finite.
* If is a finitely additive, monotone, and locally finite then is necessarily finite for every compact measurable subset
- if whenever is
directed
Direct may refer to:
Mathematics
* Directed set, in order theory
* Direct limit of (pre), sheaves
* Direct sum of modules, a construction in abstract algebra which combines several vector spaces
Computing
* Direct access (disambiguation), a ...
with respect to and satisfies
* is directed
Direct may refer to:
Mathematics
* Directed set, in order theory
* Direct limit of (pre), sheaves
* Direct sum of modules, a construction in abstract algebra which combines several vector spaces
Computing
* Direct access (disambiguation), a ...
with respect to if and only if it is not empty and for all there exists some such that and
- or if for every
- if for every
- if it is both inner regular and outer regular.
- a if it is a Borel measure that is also .
- a if it is a regular and locally finite measure.
- if every non-empty open subset has (strictly) positive measure.
- a if it is non-negative, monotone, modular, has a null empty set, and has domain
Relationships between set functions
If and are two set functions over then:
- is said to be or , written if for every set that belongs to the domain of both and if then
* If and are -finite measures on the same measurable space and if then the Radon–Nikodym derivative exists and for every measurable
* and are called if each one is absolutely continuous with respect to the other. is called a of a measure if is -finite and they are equivalent.
- and are , written if there exist disjoint sets and in the domains of and such that for all in the domain of and for all in the domain of
Examples
Examples of set functions include:
* The function assigning densities to sufficiently well-behaved
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved or n ...
subsets is a set function.
* A probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
assigns a probability to each set in a σ-algebra. Specifically, the probability of the empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
is zero and the probability of the sample space is with other sets given probabilities between and
* A possibility measure assigns a number between zero and one to each set in the powerset of some given set. See possibility theory.
* A is a set-valued random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
. See the article random compact set.
The Jordan measure on is a set function defined on the set of all Jordan measurable subsets of it sends a Jordan measurable set to its Jordan measure.
Lebesgue measure
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 higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
on is a set function that assigns a non-negative real number to every set of real numbers that belongs to the Lebesgue -algebra.[Kolmogorov and Fomin 1975]
Its definition begins with the set of all intervals of real numbers, which is a semialgebra on
The function that assigns to every interval its is a finitely additive set function (explicitly, if has endpoints then ).
This set function can be extended to the Lebesgue outer measure on which is the translation-invariant set function 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 higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
.
Vitali sets are examples of non-measurable sets of real numbers.
Infinite-dimensional space
As detailed in the article on infinite-dimensional Lebesgue measure, the only locally finite and translation-invariant 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.
...
on an infinite-dimensional separable normed space is the trivial measure. However, it is possible to define Gaussian measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space \mathbb^n, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are na ...
s on infinite-dimensional topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s. The structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
.
Finitely additive translation-invariant set functions
The only translation-invariant measure on \Omega = \Reals with domain \wp(\Reals) that is finite on every compact subset of \Reals is the trivial set function \wp(\Reals) \to , \infty/math> that is identically equal to 0 (that is, it sends every S \subseteq \Reals to 0)
However, if countable additivity is weakened to finite additivity then a non-trivial set function with these properties does exist and moreover, some are even valued in , 1 In fact, such non-trivial set functions will exist even if \Reals is replaced by any other abelian group G.
Extending set functions
Extending from semialgebras to algebras
Suppose that \mu is a set function on a semialgebra \mathcal over \Omega and let
\operatorname(\mathcal) := \left\,
which is the algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
on \Omega generated by \mathcal.
The archetypal
The concept of an archetype ( ) appears in areas relating to behavior, History of psychology#Emergence of German experimental psychology, historical psychology, philosophy and literary analysis.
An archetype can be any of the following:
# a stat ...
example of a semialgebra that is not also an algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
is the family
\mathcal_d := \ \cup \left\
on \Omega := \R^d where (a, b] := \ for all -\infty \leq a < b \leq \infty. Importantly, the two non-strict inequalities \,\leq\, in -\infty \leq a_i < b_i \leq \infty cannot be replaced with strict inequalities \,<\, since semialgebras must contain the whole underlying set \R^d; that is, \R^d \in \mathcal_d is a requirement of semialgebras (as is \varnothing \in \mathcal_d).
If \mu is finitely additive then it has a unique extension to a set function \overline on \operatorname(\mathcal) defined by sending F_1 \sqcup \cdots \sqcup F_n \in \operatorname(\mathcal) (where \,\sqcup\, indicates that these F_i \in \mathcal are pairwise disjoint
In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...
) to:
\overline\left(F_1 \sqcup \cdots \sqcup F_n\right) := \mu\left(F_1\right) + \cdots + \mu\left(F_n\right).
This extension \overline will also be finitely additive: for any pairwise disjoint A_1, \ldots, A_n \in \operatorname(\mathcal),
\overline\left(A_1 \cup \cdots \cup A_n\right) = \overline\left(A_1\right) + \cdots + \overline\left(A_n\right).
If in addition \mu is extended real-valued and monotone (which, in particular, will be the case if \mu is non-negative) then \overline will be monotone and finitely subadditive: for any A, A_1, \ldots, A_n \in \operatorname(\mathcal) such that A \subseteq A_1 \cup \cdots \cup A_n,
\overline\left(A\right) \leq \overline\left(A_1\right) + \cdots + \overline\left(A_n\right).
Extending from rings to σ-algebras
If \mu : \mathcal \to , \infty/math> is a pre-measure on a ring of sets (such as an algebra of sets) \mathcal over \Omega then \mu has an extension to a measure \overline : \sigma(\mathcal) \to , \infty/math> on the σ-algebra \sigma(\mathcal) generated by \mathcal. If \mu is σ-finite then this extension is unique.
To define this extension, first extend \mu to an outer measure \mu^* on 2^\Omega = \wp(\Omega) by
\mu^*(T) = \inf \left\
and then restrict it to the set \mathcal_M of \mu^*-measurable sets (that is, Carathéodory-measurable sets), which is the set of all M \subseteq \Omega such that
\mu^*(S) = \mu^*(S \cap M) + \mu^*(S \cap M^\mathrm) \quad \text S \subseteq \Omega. It is a \sigma-algebra and \mu^* is sigma-additive on it, by Caratheodory lemma.
Restricting outer measures
If \mu^* : \wp(\Omega) \to , \infty/math> is an outer measure on a set \Omega, where (by definition) the domain is necessarily the power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
\wp(\Omega) of \Omega, then a subset M \subseteq \Omega is called or if it satisfies the following :
\mu^*(S) = \mu^*(S \cap M) + \mu^*(S \cap M^\mathrm) \quad \text S \subseteq \Omega,
where M^\mathrm := \Omega \setminus M is the complement of M.
The family of all \mu^*–measurable subsets is a σ-algebra and the restriction of the outer measure \mu^* to this family is a measure.
See also
*
*
*
*
*
*
*
*
*
*
*
*
*
*
Notes
Proofs
References
*
*
* A. N. Kolmogorov and S. V. Fomin (1975), ''Introductory Real Analysis'', Dover.
*
*
Further reading
*
Regular set function
a
Encyclopedia of Mathematics
{{Analysis in topological vector spaces
Basic concepts in set theory
Functions and mappings
Measure theory
Measures (measure theory)