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) | Never | |||||||||
Ξ΄-Ring | Never | |||||||||
π-Ring | Never | |||||||||
Algebra (Field) | Never | |||||||||
π-Algebra (π-Field) | Never | |||||||||
Dual ideal | ||||||||||
Filter | Never | Never | ||||||||
Prefilter (Filter base) | Never | Never | ||||||||
Filter subbase | Never | Never | ||||||||
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 |
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) | Never | |||||||||
Ξ΄-Ring | Never | |||||||||
π-Ring | Never | |||||||||
Algebra (Field) | Never | |||||||||
π-Algebra (π-Field) | Never | |||||||||
Dual ideal | ||||||||||
Filter | Never | Never | ||||||||
Prefilter (Filter base) | Never | Never | ||||||||
Filter subbase | Never | Never | ||||||||
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 |