Sigma-ring
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.
Formal definition
[edit]Let be a nonempty collection of sets. Then is a π-ring if:
- Closed under countable unions: if for all
- Closed under relative complementation: if
Properties
[edit]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.
Similar concepts
[edit]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.
Uses
[edit]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
See also
[edit]- δ-ring β Ring closed under countable intersections
- Field of sets β Algebraic concept in measure theory, also referred to as an algebra of sets
- Join (sigma algebra) β Algebraic structure of set algebra
- π-system (Dynkin system) β Family closed under complements and countable disjoint unions
- Measurable function β Function for which the preimage of a measurable set is measurable
- Monotone class β theorem
- Ο-system β Family of sets closed under intersection
- Ring of sets β Family closed under unions and relative complements
- Sample space β Set of all possible outcomes or results of a statistical trial or experiment
- π additivity β Mapping function
- Ο-algebra β Algebraic structure of set algebra
- π-ideal β Family closed under subsets and countable unions
References
[edit]- Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses π-rings in development of Lebesgue theory.
Families of sets over | ||||||||||
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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 |