In mathematics, a nonempty collection of sets is called a π-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.
Let be a nonempty collection of sets. Then is a π-ring if:
These two properties imply: whenever are elements of
This is because
Every π-ring is a Ξ΄-ring but there exist Ξ΄-rings that are not π-rings.
If the first property is weakened to closure under finite union (that is, whenever ) but not countable union, then is a ring but not a π-ring.
π-rings can be used instead of π-fields (π-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every π-field is also a π-ring, but a π-ring need not be a π-field.
A π-ring that is a collection of subsets of induces a π-field for Define Then is a π-field over the set - to check closure under countable union, recall a -ring is closed under countable intersections. In fact is the minimal π-field containing since it must be contained in every π-field containing
Families of sets over | ||||||||||
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Is necessarily true of or, is closed under: |
Directed by |
F.I.P. | ||||||||
Ο-system | ![]() |
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Semiring | ![]() |
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Never |
Semialgebra (Semifield) | ![]() |
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Never |
Monotone class | ![]() |
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only if | only if | ![]() |
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π-system (Dynkin System) | ![]() |
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only if |
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only if or they are disjoint |
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Never |
Ring (Order theory) | ![]() |
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Ring (Measure theory) | ![]() |
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Never |
Ξ΄-Ring | ![]() |
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Never |
π-Ring | ![]() |
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Never |
Algebra (Field) | ![]() |
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Never |
π-Algebra (π-Field) | ![]() |
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Never |
Dual ideal | ![]() |
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Filter | ![]() |
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Never | Never | ![]() |
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Prefilter (Filter base) | ![]() |
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Never | Never | ![]() |
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Filter subbase | ![]() |
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Never | Never | ![]() |
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Open Topology | ![]() |
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![]() (even arbitrary ) |
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Never |
Closed Topology | ![]() |
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![]() (even arbitrary ) |
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Never |
Is necessarily true of 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
Ο-system where every complement is equal to a finite
disjoint union of sets in |
In mathematics, a nonempty collection of sets is called a π-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.
Let be a nonempty collection of sets. Then is a π-ring if:
These two properties imply: whenever are elements of
This is because
Every π-ring is a Ξ΄-ring but there exist Ξ΄-rings that are not π-rings.
If the first property is weakened to closure under finite union (that is, whenever ) but not countable union, then is a ring but not a π-ring.
π-rings can be used instead of π-fields (π-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every π-field is also a π-ring, but a π-ring need not be a π-field.
A π-ring that is a collection of subsets of induces a π-field for Define Then is a π-field over the set - to check closure under countable union, recall a -ring is closed under countable intersections. In fact is the minimal π-field containing since it must be contained in every π-field containing
Families of sets over | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Is necessarily true of or, is closed under: |
Directed by |
F.I.P. | ||||||||
Ο-system | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Semiring | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Never |
Semialgebra (Semifield) | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Never |
Monotone class | ![]() |
![]() |
![]() |
![]() |
![]() |
only if | only if | ![]() |
![]() |
![]() |
π-system (Dynkin System) | ![]() |
![]() |
![]() |
only if |
![]() |
![]() |
only if or they are disjoint |
![]() |
![]() |
Never |
Ring (Order theory) | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Ring (Measure theory) | ![]() |
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Never |
Ξ΄-Ring | ![]() |
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Never |
π-Ring | ![]() |
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![]() |
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![]() |
![]() |
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Never |
Algebra (Field) | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Never |
π-Algebra (π-Field) | ![]() |
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![]() |
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![]() |
![]() |
![]() |
Never |
Dual ideal | ![]() |
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![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
Filter | ![]() |
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Never | Never | ![]() |
![]() |
![]() |
![]() | |
Prefilter (Filter base) | ![]() |
![]() |
![]() |
Never | Never | ![]() |
![]() |
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Filter subbase | ![]() |
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![]() |
Never | Never | ![]() |
![]() |
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![]() | |
Open Topology | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() (even arbitrary ) |
![]() |
![]() |
Never |
Closed Topology | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() (even arbitrary ) |
![]() |
![]() |
![]() |
Never |
Is necessarily true of 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
Ο-system where every complement is equal to a finite
disjoint union of sets in |