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Sigma-ring
Family of sets closed under countable unions From Wikipedia, the free encyclopedia
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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
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
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Properties
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Perspective
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.
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Similar concepts
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
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Perspective
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
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See also
- Ξ΄-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 Kind of mathematical function
- Monotone class β Measure theory and probability 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
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References
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