Function from sets to numbers
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line
which consists of the real numbers
and
A set function generally aims to measure 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 over
(meaning that
where
denotes the powerset) then a set function on
is a function
with domain
and codomain
or, sometimes, the codomain is instead some vector space, 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.
Families of sets over |
Is necessarily true of ![{\displaystyle {\mathcal {F}}\colon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c806bc7022198fb7b8ddd4a0b376329bb77e00c) or, is closed under: | Directed by | | | | | | | | | F.I.P. |
Ï-system | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | |
Semiring | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | Never |
Semialgebra (Semifield) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | Never |
Monotone class | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | only if ![{\displaystyle A_{i}\searrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba4f0f9c907ac9321bf8494f69cc190cbf8a56d) | only if ![{\displaystyle A_{i}\nearrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b851ff0dcb2264bbedafbef85a71e4f98c842865) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | |
đ-system (Dynkin System) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | only if
![{\displaystyle A\subseteq B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b09068bd2f7ba899aeb883ebe670b2ad07b0c851) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | only if or they are disjoint | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | Never |
Ring (Order theory) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | |
Ring (Measure theory) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | Never |
ÎŽ-Ring | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | Never |
đ-Ring | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | Never |
Algebra (Field) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | Never |
đ-Algebra (đ-Field) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | Never |
Dual ideal | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | |
Filter | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | Never | Never | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![{\displaystyle \varnothing \not \in {\mathcal {F}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c99d2db231b6a5af19206e95ff6d98d3019e9b) | |
Prefilter (Filter base) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | Never | Never | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![{\displaystyle \varnothing \not \in {\mathcal {F}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c99d2db231b6a5af19206e95ff6d98d3019e9b) | |
Filter subbase | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | Never | Never | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![{\displaystyle \varnothing \not \in {\mathcal {F}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c99d2db231b6a5af19206e95ff6d98d3019e9b) | |
Open Topology | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![](//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png) (even arbitrary ) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | Never |
Closed Topology | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![](//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png) (even arbitrary ) | ![No](//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | ![Yes](//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) | Never |
Is necessarily true of ![{\displaystyle {\mathcal {F}}\colon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c806bc7022198fb7b8ddd4a0b376329bb77e00c) or, is closed under: | directed downward | finite intersections | finite unions | relative complements | complements in | countable intersections | countable unions | contains | contains | Finite Intersection Property |
Additionally, a semiring is a Ï-system where every complement is equal to a finite disjoint union of sets in ![{\displaystyle {\mathcal {F}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1656ae73ede684468b360e948a8a38e6e2c461) A semialgebra is a semiring where every complement is equal to a finite disjoint union of sets in ![{\displaystyle {\mathcal {F}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1656ae73ede684468b360e948a8a38e6e2c461) are arbitrary elements of and it is assumed that ![{\displaystyle {\mathcal {F}}\neq \varnothing .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ed685bdf4c75742b28ccec093cae48329c1a9d6) |
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:
- Set difference formula:
is defined with
satisfying
and ![{\displaystyle F\setminus E\in {\mathcal {F}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33e65bdeac5e8ed24a92796492e36da79387f347)
Null sets
A set
is called a null set (with respect to
) or simply null if
Whenever
is not identically equal to either
or
then it is typically also assumed that:
- null empty set:
if ![{\displaystyle \varnothing \in {\mathcal {F}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f37264659309ae1379415c23f13d833d4cd5b7)
Variation and mass
The total variation of a set
is
where
denotes the absolute value (or more generally, it denotes the norm or seminorm if
is vector-valued in a (semi)normed space). Assuming that
then
is called the total variation of
and
is called the mass of
A set function is called finite if for every
the value
is finite (which by definition means that
and
; an infinite value is one that is equal to
or
). Every finite set function must have a finite mass.
Common properties of set functions
A set function
on
is said to be
- non-negative if it is valued in
![{\displaystyle [0,\infty ].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f850b8f222b718ddb41c8163d1a995e461126c37)
- finitely additive if
for all pairwise disjoint finite sequences
such that
- If
is closed under binary unions then
is finitely additive if and only if
for all disjoint pairs ![{\displaystyle E,F\in {\mathcal {F}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77c0f35b3881ff0b4849d58e6265a99ae93d3c07)
- 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).
- countably additive or Ï-additive if in addition to being finitely additive, for all pairwise disjoint 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
![{\displaystyle \textstyle \sum \limits _{i=1}^{\infty }\mu \left(F_{i}\right)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\displaystyle \lim _{n\to \infty }}\mu \left(F_{1}\right)+\cdots +\mu \left(F_{n}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aabd6c0f4007358a19543e0d5c64f692ed933ad4)
- As a consequence, if
is any permutation/bijection 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 ![{\displaystyle \mu (F)=\mu \left(F_{\rho (1)}\right)+\mu \left(F_{\rho (2)}\right)+\cdots \,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afa8091db15d1ecf8acda312f81291c3fb515922)
- 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 ![{\displaystyle [-\infty ,\infty ].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7c53f28e818adb1f7baadb9f66eb45aebc1c33d)
- 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 pre-measure if it is non-negative, countably additive (including finitely additive), and has a null empty set.
- a measure if it is a pre-measure whose domain is a Ï-algebra. That is to say, a measure is a non-negative countably additive set function on a Ï-algebra that has a null empty set.
- a probability measure if it is a measure that has a mass of
![{\displaystyle 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af8c4e445819b13a052647aa3eb2be990b0a4b24)
- an outer measure if it is non-negative, countably subadditive, has a null empty set, and has the power set
as its domain. - a signed measure if it is countably additive, has a null empty set, and
does not take on both
and
as values. - complete 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).
- đ-finite if there exists a sequence
in
such that
is finite for every index
and also ![{\displaystyle \textstyle \bigcup \limits _{n=1}^{\infty }F_{n}=\textstyle \bigcup \limits _{F\in {\mathcal {F}}}F.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6070d189e049746ee06a9b092cd804ea885da0)
- decomposable if there exists a subfamily
of pairwise disjoint sets such that
is finite for every
and also
(where
). - Every đ-finite set function is decomposable although not conversely. For example, the counting measure on
(whose domain is
) is decomposable but not đ-finite.
- a vector measure if it is a countably additive set function
valued in a topological vector space
(such as a normed space) whose domain is a Ï-algebra. - If
is valued in a normed space
then it is countably additive if and only if for any pairwise disjoint sequence
in
If
is finitely additive and valued in a Banach space then it is countably additive if and only if for any pairwise disjoint sequence
in
![{\displaystyle \lim _{n\to \infty }\left\|\mu \left(F_{n}\cup F_{n+1}\cup F_{n+2}\cup \cdots \right)\right\|=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/990ad5d4a0f72c6d4448b1d37f53271d6df12320)
- a complex measure if it is a countably additive complex-valued set function
whose domain is a Ï-algebra. - By definition, a complex measure never takes
as a value and so has a null empty set.
- a random measure 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 numbers 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 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 (that is, either finite or countably infinite); this remains true if
is replaced with any normed space.[proof 1] 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
- monotone if
whenever
satisfy ![{\displaystyle E\subseteq F.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c43b37bd26bea5a94a90bdfa416a873b8574c36d)
- modular if it satisfies the following condition, known as modularity:
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 valuation. This geometric definition of "valuation" should not be confused with the stronger non-equivalent measure theoretic definition of "valuation" that is given below.
- submodular if
for all
such that ![{\displaystyle E\cup F,E\cap F\in {\mathcal {F}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c866026b69b87a6264f454ce07ce44adb78363e8)
- finitely subadditive if
for all finite sequences
that satisfy ![{\displaystyle F\;\subseteq \;\textstyle \bigcup \limits _{i=1}^{n}F_{i}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/126273370d5c09dae346e31acae286cf9e1c370a)
- countably subadditive or Ï-subadditive 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 then this inequality is Boole's inequality. - If
is countably subadditive and
with
then
is finitely subadditive.
- superadditive if
whenever
are disjoint with ![{\displaystyle E\cup F\in {\mathcal {F}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e835d543ba4263053be0c9bdb1b44ce6d05b0106)
- continuous from above if
for all non-increasing sequences 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 ![{\displaystyle \textstyle \bigcap \limits _{i=1}^{\infty }F_{i}=\varnothing .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5465ac68fdca84573d99e970eedcdef44a813ea0)
- continuous from below if
for all non-decreasing sequences of sets
in
such that ![{\displaystyle \textstyle \bigcup \limits _{i=1}^{\infty }F_{i}\in {\mathcal {F}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b91226cc8688fcedc55aa8c65ba9aff82ef355b)
- infinity is approached from below if whenever
satisfies
then for every real
there exists some
such that
and ![{\displaystyle r\leq \mu \left(F_{r}\right)<\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/caa0b0155e65799bf294dd6041724e0f9b584c2e)
- an outer measure if
is non-negative, countably subadditive, has a null empty set, and has the power set
as its domain. - an inner measure if
is non-negative, superadditive, continuous from above, has a null empty set, has the power set
as its domain, and
is approached from below. - atomic if every measurable set of positive measure contains an atom.
If a binary operation
is defined, then a set function
is said to be
- translation invariant if
for all
and
such that ![{\displaystyle \omega +F\in {\mathcal {F}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ade00a894ff1c03e35ce9e4d481b64956cd9ad50)
Topology related definitions
If
is a topology on
then a set function
is said to be:
- a Borel measure if it is a measure defined on the Ï-algebra of all Borel sets, which is the smallest Ï-algebra containing all open subsets (that is, containing
). - a Baire measure if it is a measure defined on the Ï-algebra of all Baire sets.
- locally finite 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 ![{\displaystyle K.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb0e178e42abf16ef4e4c0b0f22aa235ad6e6e5)
-additive if
whenever
is directed with respect to
and satisfies
is directed with respect to
if and only if it is not empty and for all
there exists some
such that
and ![{\displaystyle B\subseteq C.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d17fc60cbf9fd462e651ffb004a24bd93c6aad5b)
- inner regular or tight if for every
![{\displaystyle \mu (F)=\sup\{\mu (K):F\supseteq K{\text{ with }}K\in {\mathcal {F}}{\text{ a compact subset of }}(\Omega ,\tau )\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d7d7101b0e9cc4fcdd955988675521c86010840)
- outer regular if for every
![{\displaystyle \mu (F)=\inf\{\mu (U):F\subseteq U{\text{ and }}U\in {\mathcal {F}}\cap \tau \}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/389e776831d34f330cf448eff1e51bef9a8f5080)
- regular if it is both inner regular and outer regular.
- a Borel regular measure if it is a Borel measure that is also regular.
- a Radon measure if it is a regular and locally finite measure.
- strictly positive if every non-empty open subset has (strictly) positive measure.
- a valuation if it is non-negative, monotone, modular, has a null empty set, and has domain
![{\displaystyle \tau .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/871bb01391136d3551c8ea59059e106be2a403cd)
Relationships between set functions
If
and
are two set functions over
then:
is said to be absolutely continuous with respect to
or dominated by
, 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 RadonâNikodym derivative
exists and for every measurable
![{\displaystyle \mu (F)=\int _{F}{\frac {d\mu }{d\nu }}d\nu .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12111d7e99285cd998ae5dbaaf4179666b580b75)
and
are called equivalent if each one is absolutely continuous with respect to the other.
is called a supporting measure of a measure
if
is
-finite and they are equivalent.[4]
and
are singular, 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 ![{\displaystyle \nu .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12bafc5c30bf9727eb4c005ff0c0632d555b3b7d)
Examples
Examples of set functions include:
- The function
assigning densities to sufficiently well-behaved subsets
is a set function. - A probability measure assigns a probability to each set in a Ï-algebra. Specifically, the probability of the empty set is zero and the probability of the sample space is
with other sets given probabilities between
and ![{\displaystyle 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af8c4e445819b13a052647aa3eb2be990b0a4b24)
- A possibility measure assigns a number between zero and one to each set in the powerset of some given set. See possibility theory.
- A random set is a set-valued random variable. 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 on
is a set function that assigns a non-negative real number to every set of real numbers that belongs to the Lebesgue
-algebra.[5]
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
that sends a subset
to the infimum
Lebesgue outer measure is not countably additive (and so is not a measure) although its restriction to the đ-algebra of all subsets
that satisfy the Carathéodory criterion:
is a measure that called Lebesgue measure. 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 on an infinite-dimensional separable normed space is the trivial measure. However, it is possible to define Gaussian measures on infinite-dimensional topological vector spaces. 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.
Finitely additive translation-invariant set functions
The only translation-invariant measure on
with domain
that is finite on every compact subset of
is the trivial set function
that is identically equal to
(that is, it sends every
to
) 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
In fact, such non-trivial set functions will exist even if
is replaced by any other abelian group
Extending set functions
Extending from semialgebras to algebras
Suppose that
is a set function on a semialgebra
over
and let
which is the algebra on
generated by
The archetypal example of a semialgebra that is not also an algebra is the family
on
where
for all
Importantly, the two non-strict inequalities
in
cannot be replaced with strict inequalities
since semialgebras must contain the whole underlying set
that is,
is a requirement of semialgebras (as is
).
If
is finitely additive then it has a unique extension to a set function
on
defined by sending
(where
indicates that these
are pairwise disjoint) to:
This extension
will also be finitely additive: for any pairwise disjoint
If in addition
is extended real-valued and monotone (which, in particular, will be the case if
is non-negative) then
will be monotone and finitely subadditive: for any
such that
Extending from rings to Ï-algebras
If
is a pre-measure on a ring of sets (such as an algebra of sets)
over
then
has an extension to a measure
on the Ï-algebra
generated by
If
is Ï-finite then this extension is unique.
To define this extension, first extend
to an outer measure
on
by
and then restrict it to the set
of
-measurable sets (that is, Carathéodory-measurable sets), which is the set of all
such that
It is a
-algebra and
is sigma-additive on it, by Caratheodory lemma.
Restricting outer measures
If
is an outer measure on a set
where (by definition) the domain is necessarily the power set
of
then a subset
is called
âmeasurable or CarathĂ©odory-measurable if it satisfies the following CarathĂ©odory's criterion:
where
is the complement of
The family of all
âmeasurable subsets is a Ï-algebra and the restriction of the outer measure
to this family is a measure.
See also
Notes
- ^ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 21. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.
- ^ Kolmogorov and Fomin 1975
- ^ The function
being translation-invariant means that
for every
and every subset
Proofs
- ^ Suppose the net
converges to some point in a metrizable topological vector space
(such as
or a normed space), where recall that this net's domain is the directed set
Like every convergent net, this convergent net of partial sums
is a Cauchy net, which for this particular net means (by definition) that for every neighborhood
of the origin in
there exists a finite subset
of
such that
for all finite supersets
this implies that
for every
(by taking
and
). Since
is metrizable, it has a countable neighborhood basis
at the origin, whose intersection is necessarily
(since
is a Hausdorff TVS). For every positive integer
pick a finite subset
such that
for every
If
belongs to
then
belongs to
Thus
for every index
that does not belong to the countable set
References
- Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020.
- Kolmogorov, Andrey; Fomin, Sergei V. (1957). Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics. New York: Dover Books. ISBN 978-1-61427-304-2. OCLC 912495626.
- A. N. Kolmogorov and S. V. Fomin (1975), Introductory Real Analysis, Dover. ISBN 0-486-61226-0
- Royden, Halsey; Fitzpatrick, Patrick (15 January 2010). Real Analysis (4 ed.). Boston: Prentice Hall. ISBN 978-0-13-143747-0. OCLC 456836719.
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
Further reading
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Basic concepts | |
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Sets | |
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Types of Measures | |
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Particular measures | |
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Maps | |
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Main results | |
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Other results | |
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Applications & related | |
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