Family closed under complements and countable disjoint unions
A Dynkin system,[1] named after Eugene Dynkin, is a collection of subsets of another universal set
satisfying a set of axioms weaker than those of 𝜎-algebra. Dynkin systems are sometimes referred to as 𝜆-systems (Dynkin himself used this term) or d-system.[2] These set families have applications in measure theory and probability.
A major application of 𝜆-systems is the π-𝜆 theorem, see below.
Definition
Let
be a nonempty set, and let
be a collection of subsets of
(that is,
is a subset of the power set of
). Then
is a Dynkin system if
![{\displaystyle \Omega \in D;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1591a4abb374c33f5124647dbbcf6fba1b26c6cc)
is closed under complements of subsets in supersets: if
and
then ![{\displaystyle B\setminus A\in D;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c4bf351aa176243ca7aad146307b8a91e2f409d)
is closed under countable increasing unions: if
is an increasing sequence[note 1] of sets in
then ![{\displaystyle \bigcup _{n=1}^{\infty }A_{n}\in D.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/416f0ae523bdee76eedebbd5b5ca4ae321a466ba)
It is easy to check[proof 1] that any Dynkin system
satisfies:
![{\displaystyle \varnothing \in D;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67c2cfbe657acfdd208870d2081cd57aa7f49f04)
is closed under complements in
: if
then
- Taking
shows that ![{\displaystyle \varnothing \in D.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d68e8bb42a80f6ed80008ea723c80840e357801)
is closed under countable unions of pairwise disjoint sets: if
is a sequence of pairwise disjoint sets in
(meaning that
for all
) then
- To be clear, this property also holds for finite sequences
of pairwise disjoint sets (by letting
for all
).
Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class.[proof 2] For this reason, a small group of authors have adopted conditions 4-6 to define a Dynkin system.
An important fact is that any Dynkin system that is also a π-system (that is, closed under finite intersections) is a 𝜎-algebra. This can be verified by noting that conditions 2 and 3 together with closure under finite intersections imply closure under finite unions, which in turn implies closure under countable unions.
Given any collection
of subsets of
there exists a unique Dynkin system denoted
which is minimal with respect to containing
That is, if
is any Dynkin system containing
then
is called the Dynkin system generated by
For instance,
For another example, let
and
; then
Sierpiński–Dynkin's π-λ theorem
Sierpiński-Dynkin's π-𝜆 theorem:[3] If
is a π-system and
is a Dynkin system with
then
In other words, the 𝜎-algebra generated by
is contained in
Thus a Dynkin system contains a π-system if and only if it contains the 𝜎-algebra generated by that π-system.
One application of Sierpiński-Dynkin's π-𝜆 theorem is the uniqueness of a measure that evaluates the length of an interval (known as the Lebesgue measure):
Let
be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let
be another measure on
satisfying
and let
be the family of sets
such that
Let
and observe that
is closed under finite intersections, that
and that
is the 𝜎-algebra generated by
It may be shown that
satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's π-𝜆 Theorem it follows that
in fact includes all of
, which is equivalent to showing that the Lebesgue measure is unique on
.
Application to probability distributions
This section is transcluded from pi system. (edit | history)
The π-𝜆 theorem motivates the common definition of the probability distribution of a random variable
in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable is defined as
![{\displaystyle F_{X}(a)=\operatorname {P} [X\leq a],\qquad a\in \mathbb {R} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce7f397fb0c38c13d07c65d9fcf82497904cef89)
whereas the seemingly more general
law of the variable is the probability measure
![{\displaystyle {\mathcal {L}}_{X}(B)=\operatorname {P} \left[X^{-1}(B)\right]\quad {\text{ for all }}B\in {\mathcal {B}}(\mathbb {R} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37a28bbc802a4b857d4f7433d2d3861de4de4c51)
where
![{\displaystyle {\mathcal {B}}(\mathbb {R} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b72c5154b8532f1f97a9d217a1ec867e934e772f)
is the Borel 𝜎-algebra. The random variables
![{\displaystyle X:(\Omega ,{\mathcal {F}},\operatorname {P} )\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac31cbd71d7abc2e9d878e4d8126e77d39e256e)
and
![{\displaystyle Y:({\tilde {\Omega }},{\tilde {\mathcal {F}}},{\tilde {\operatorname {P} }})\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b688a4cac5c32b2c5d535f5d879a84488c4b7fe9)
(on two possibly different probability spaces) are
equal in distribution (or
law), denoted by
![{\displaystyle X\,{\stackrel {\mathcal {D}}{=}}\,Y,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1eaff100592c0641246d54b87ca6e34dc49eaa8)
if they have the same cumulative distribution functions; that is, if
![{\displaystyle F_{X}=F_{Y}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a887f4cd5d2faa8195593f7a10c39500899ba653)
The motivation for the definition stems from the observation that if
![{\displaystyle F_{X}=F_{Y},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c90dc9e02c827e8cfb76b71032e2b666e41b49)
then that is exactly to say that
![{\displaystyle {\mathcal {L}}_{X}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce4db2d17b365a7321dbfdb8f8bc512dd911ea54)
and
![{\displaystyle {\mathcal {L}}_{Y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c037c65fe6ca9b086552f1c8b2b8d027a042e08)
agree on the
π-system
![{\displaystyle \{(-\infty ,a]:a\in \mathbb {R} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a208e779fb539e04e1d1e05d9bc26d34359f8ab0)
which generates
![{\displaystyle {\mathcal {B}}(\mathbb {R} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5155ab2ab5020e03bffef1ce38bb3fccfcd380)
and so by the
example above:
A similar result holds for the joint distribution of a random vector. For example, suppose
and
are two random variables defined on the same probability space
with respectively generated π-systems
and
The joint cumulative distribution function of
is
![{\displaystyle F_{X,Y}(a,b)=\operatorname {P} [X\leq a,Y\leq b]=\operatorname {P} \left[X^{-1}((-\infty ,a])\cap Y^{-1}((-\infty ,b])\right],\quad {\text{ for all }}a,b\in \mathbb {R} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a2b7fac21544d1da36d39beccfa789d7138083)
However,
and
Because
![{\displaystyle {\mathcal {I}}_{X,Y}=\left\{A\cap B:A\in {\mathcal {I}}_{X},{\text{ and }}B\in {\mathcal {I}}_{Y}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd007253f5d59913b6998701931638536b994f54)
is a
π-system generated by the random pair
![{\displaystyle (X,Y),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d7292c246e4e1e0da3897a30b078724dcc4a7fe)
the
π-𝜆 theorem is used to show that the joint cumulative distribution function suffices to determine the joint law of
![{\displaystyle (X,Y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e25a265419fe85cbbd9c050d52acfa17c907330f)
In other words,
![{\displaystyle (X,Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41f29b9537685f499713112d6802e811cbf51bba)
and
![{\displaystyle (W,Z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d1eed88de423e538890da4178eec3b57ecebde)
have the same distribution if and only if they have the same joint cumulative distribution function.
In the theory of stochastic processes, two processes
are known to be equal in distribution if and only if they agree on all finite-dimensional distributions; that is, for all
![{\displaystyle \left(X_{t_{1}},\ldots ,X_{t_{n}}\right)\,{\stackrel {\mathcal {D}}{=}}\,\left(Y_{t_{1}},\ldots ,Y_{t_{n}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d39e3036e32b1a78feac8c75720f3b178606fc99)
The proof of this is another application of the π-𝜆 theorem.[4]
See also
- Algebra of sets – Identities and relationships involving sets
- δ-ring – Ring closed under countable intersections
- Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
- Monotone class – theoremPages displaying wikidata descriptions as a fallbackPages displaying short descriptions with no spaces
- π-system – Family of sets closed under intersection
- Ring of sets – Family closed under unions and relative complements
- σ-algebra – Algebraic structure of set algebra
- 𝜎-ideal – Family closed under subsets and countable unions
- 𝜎-ring – Ring closed under countable unions
Notes
- ^ A sequence of sets
is called increasing if
for all
Proofs
- ^ Assume
satisfies (1), (2), and (3). Proof of (5) :Property (5) follows from (1) and (2) by using
The following lemma will be used to prove (6). Lemma: If
are disjoint then
Proof of Lemma:
implies
where
by (5). Now (2) implies that
contains
so that (5) guarantees that
which proves the lemma. Proof of (6) Assume that
are pairwise disjoint sets in
For every integer
the lemma implies that
where because
is increasing, (3) guarantees that
contains their union
as desired.
- ^ Assume
satisfies (4), (5), and (6). proof of (2): If
satisfy
then (5) implies
and since
(6) implies that
contains
so that finally (4) guarantees that
is in
Proof of (3): Assume
is an increasing sequence of subsets in
let
and let
for every
where (2) guarantees that
all belong to
Since
are pairwise disjoint, (6) guarantees that their union
belongs to
which proves (3).
- ^ Dynkin, E., "Foundations of the Theory of Markov Processes", Moscow, 1959
- ^ Aliprantis, Charalambos; Border, Kim C. (2006). Infinite Dimensional Analysis: a Hitchhiker's Guide (Third ed.). Springer. Retrieved August 23, 2010.
- ^ Sengupta. "Lectures on measure theory lecture 6: The Dynkin π − λ Theorem" (PDF). Math.lsu. Retrieved 3 January 2023.
- ^ Kallenberg, Foundations Of Modern Probability, p. 48
References
- Gut, Allan (2005). Probability: A Graduate Course. New York: Springer. doi:10.1007/b138932. ISBN 0-387-22833-0.
- Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. ISBN 0-471-00710-2.
- Williams, David (2007). Probability with Martingales. Cambridge University Press. p. 193. ISBN 0-521-40605-6.
This article incorporates material from Dynkin system on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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. |
π-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) | |
𝜆-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 |
δ-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 |
𝜎-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 |
𝜎-Algebra (𝜎-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 π-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) |
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Maps | |
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