Sigma-ring
In mathematics, a nonempty collection of sets is called a 𝜎-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 𝜎-ring if:
- Closed under countable unions: if for all
- Closed under relative complementation: if
Properties
These two properties imply: whenever are elements of
This is because
Every 𝜎-ring is a δ-ring but there exist δ-rings that are not 𝜎-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 𝜎-ring.
Uses
𝜎-rings can be used instead of 𝜎-fields (𝜎-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every 𝜎-field is also a 𝜎-ring, but a 𝜎-ring need not be a 𝜎-field.
A 𝜎-ring that is a collection of subsets of induces a 𝜎-field for Define Then is a 𝜎-field over the set - to check closure under countable union, recall a -ring is closed under countable intersections. In fact is the minimal 𝜎-field containing since it must be contained in every 𝜎-field containing
See also
- δ-ring – Ring closed under countable intersections
- Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
- Join (sigma algebra) – Algebraic structure of set algebraPages displaying short descriptions of redirect targets
- 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions
- Measurable function – Function for which the preimage of a measurable set is measurable
- Monotone class – theoremPages displaying wikidata descriptions as a fallbackPages displaying short descriptions with no spaces
- π-system – Family of sets closed under intersection
- Ring of sets – Family closed under unions and relative complements
- Sample space – Set of all possible outcomes or results of a statistical trial or experiment
- 𝜎 additivity – Mapping function
- σ-algebra – Algebraic structure of set algebra
- 𝜎-ideal – Family closed under subsets and countable unions
References
- Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses 𝜎-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. | ||||||||
π-system | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
Semiring | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | Never |
Semialgebra (Semifield) | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | Never |
Monotone class | ![]() | ![]() | ![]() | ![]() | ![]() | only if | only if | ![]() | ![]() | ![]() |
𝜆-system (Dynkin System) | ![]() | ![]() | ![]() | only if | ![]() | ![]() | only if or they are disjoint | ![]() | ![]() | Never |
Ring (Order theory) | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
Ring (Measure theory) | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | Never |
δ-Ring | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | Never |
𝜎-Ring | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | Never |
Algebra (Field) | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | Never |
𝜎-Algebra (𝜎-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 π-system where every complement is equal to a finite disjoint union of sets in |