Sigma-ring
In mathematics, a nonempty collection of sets is called a ๐-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 ๐-ring if:
- Closed under countable unions: if for all
- Closed under relative complementation: if
Properties
These two properties imply:
This is because
Every ๐-ring is a ฮด-ring but there exist ฮด-rings that are not ๐-rings.
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 ๐-ring.
Uses
๐-rings can be used instead of ๐-fields (๐-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every ๐-field is also a ๐-ring, but a ๐-ring need not be a ๐-field.
A ๐-ring that is a collection of subsets of induces a ๐-field for Define Then is a ๐-field over the set - to check closure under countable union, recall a -ring is closed under countable intersections. In fact is the minimal ๐-field containing since it must be contained in every ๐-field containing
See also
- ฮด-ringย โ Ring closed under countable intersections
- Field of setsย โ Algebraic concept in measure theory, also referred to as an algebra of sets
- Join (sigma algebra)ย โ Algebraic structure of set algebra
- ๐-system (Dynkin system)ย โ Family closed under complements and countable disjoint unions
- Measurable functionย โ Function for which the preimage of a measurable set is measurable
- Monotone classย โ theorem
- ฯ-systemย โ Family of sets closed under intersection
- Ring of setsย โ Family closed under unions and relative complements
- Sample spaceย โ Set of all possible outcomes or results of a statistical trial or experiment
- ๐ additivityย โ Mapping function
- ฯ-algebraย โ Algebraic structure of set algebra
- ๐-idealย โ Family closed under subsets and countable unions
References
- Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses ๐-rings in development of Lebesgue theory.
Families of sets over | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Is necessarily true of or, is closed under: |
Directed by |
F.I.P. | ||||||||
ฯ-system | ||||||||||
Semiring | Never | |||||||||
Semialgebra (Semifield) | Never | |||||||||
Monotone class | only if | only if | ||||||||
๐-system (Dynkin System) | only if |
only if or they are disjoint |
Never | |||||||
Ring (Order theory) | ||||||||||
Ring (Measure theory) | Never | |||||||||
ฮด-Ring | Never | |||||||||
๐-Ring | Never | |||||||||
Algebra (Field) | Never | |||||||||
๐-Algebra (๐-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 ฯ-system where every complement is equal to a finite disjoint union of sets in |