Sigma-ring

Ring closed under countable unions

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 R {\displaystyle {\mathcal {R}}} be a nonempty collection of sets. Then R {\displaystyle {\mathcal {R}}} is a 𝜎-ring if:

  1. Closed under countable unions: ⋃ n = 1 ∞ A n ∈ R {\displaystyle \bigcup _{n=1}^{\infty }A_{n}\in {\mathcal {R}}} if A n ∈ R {\displaystyle A_{n}\in {\mathcal {R}}} for all n ∈ N {\displaystyle n\in \mathbb {N} }
  2. Closed under relative complementation: A ∖ B ∈ R {\displaystyle A\setminus B\in {\mathcal {R}}} if A , B ∈ R {\displaystyle A,B\in {\mathcal {R}}}

Properties

These two properties imply:

⋂ n = 1 ∞ A n ∈ R {\displaystyle \bigcap _{n=1}^{\infty }A_{n}\in {\mathcal {R}}}
whenever A 1 , A 2 , … {\displaystyle A_{1},A_{2},\ldots } are elements of R . {\displaystyle {\mathcal {R}}.}

This is because

⋂ n = 1 ∞ A n = A 1 ∖ ⋃ n = 2 ∞ ( A 1 ∖ A n ) . {\displaystyle \bigcap _{n=1}^{\infty }A_{n}=A_{1}\setminus \bigcup _{n=2}^{\infty }\left(A_{1}\setminus A_{n}\right).}

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, A ∪ B ∈ R {\displaystyle A\cup B\in {\mathcal {R}}} whenever A , B ∈ R {\displaystyle A,B\in {\mathcal {R}}} ) but not countable union, then R {\displaystyle {\mathcal {R}}} 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 R {\displaystyle {\mathcal {R}}} that is a collection of subsets of X {\displaystyle X} induces a 𝜎-field for X . {\displaystyle X.} Define A = { E ⊆ X : E ∈ R   or   E c ∈ R } . {\displaystyle {\mathcal {A}}=\{E\subseteq X:E\in {\mathcal {R}}\ {\text{or}}\ E^{c}\in {\mathcal {R}}\}.} Then A {\displaystyle {\mathcal {A}}} is a 𝜎-field over the set X {\displaystyle X} - to check closure under countable union, recall a σ {\displaystyle \sigma } -ring is closed under countable intersections. In fact A {\displaystyle {\mathcal {A}}} is the minimal 𝜎-field containing R {\displaystyle {\mathcal {R}}} since it must be contained in every 𝜎-field containing R . {\displaystyle {\mathcal {R}}.}

See also

  • 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 algebraPages displaying short descriptions of redirect targets
  • πœ†-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€“ theoremPages displaying wikidata descriptions as a fallbackPages displaying short descriptions with no spaces
  • 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
  • 𝜎 additivity β€“ Mapping function
  • Οƒ-algebra β€“ Algebraic structure of set algebra
  • 𝜎-ideal β€“ 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 F {\displaystyle {\mathcal {F}}} of sets over Ω {\displaystyle \Omega }
  • v
  • t
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Is necessarily true of F : {\displaystyle {\mathcal {F}}\colon }
or, is F {\displaystyle {\mathcal {F}}} closed under: 		Directed
by ⊇ {\displaystyle \,\supseteq } 		A ∩ B {\displaystyle A\cap B} A ∪ B {\displaystyle A\cup B} B ∖ A {\displaystyle B\setminus A} Ω ∖ A {\displaystyle \Omega \setminus A} A 1 ∩ A 2 ∩ ⋯ {\displaystyle A_{1}\cap A_{2}\cap \cdots } A 1 ∪ A 2 ∪ ⋯ {\displaystyle A_{1}\cup A_{2}\cup \cdots } Ω ∈ F {\displaystyle \Omega \in {\mathcal {F}}} ∅ ∈ F {\displaystyle \varnothing \in {\mathcal {F}}} F.I.P.
Ο€-system Yes Yes No No No No No No No No
Semiring Yes Yes No No No No No No Yes Never
Semialgebra (Semifield) Yes Yes No No No No No No Yes Never
Monotone class No No No No No only if A i ↘ {\displaystyle A_{i}\searrow } only if A i ↗ {\displaystyle A_{i}\nearrow } No No No
πœ†-system (Dynkin System) Yes No No only if
A ⊆ B {\displaystyle A\subseteq B} 		Yes No only if A i ↗ {\displaystyle A_{i}\nearrow } or
they are disjoint		 Yes Yes Never
Ring (Order theory) Yes Yes Yes No No No No No No No
Ring (Measure theory) Yes Yes Yes Yes No No No No Yes Never
Ξ΄-Ring Yes Yes Yes Yes No Yes No No Yes Never
𝜎-Ring Yes Yes Yes Yes No Yes Yes No Yes Never
Algebra (Field) Yes Yes Yes Yes Yes No No Yes Yes Never
𝜎-Algebra (𝜎-Field) Yes Yes Yes Yes Yes Yes Yes Yes Yes Never
Dual ideal Yes Yes Yes No No No Yes Yes No No
Filter Yes Yes Yes Never Never No Yes Yes ∅ ∉ F {\displaystyle \varnothing \not \in {\mathcal {F}}} Yes
Prefilter (Filter base) Yes No No Never Never No No No ∅ ∉ F {\displaystyle \varnothing \not \in {\mathcal {F}}} Yes
Filter subbase No No No Never Never No No No ∅ ∉ F {\displaystyle \varnothing \not \in {\mathcal {F}}} Yes
Open Topology Yes Yes Yes No No No
(even arbitrary ∪ {\displaystyle \cup } )		 Yes Yes Never
Closed Topology Yes Yes Yes No No
(even arbitrary ∩ {\displaystyle \cap } )		 No Yes Yes Never
Is necessarily true of F : {\displaystyle {\mathcal {F}}\colon }
or, is F {\displaystyle {\mathcal {F}}} closed under: 		directed
downward 		finite
intersections 		finite
unions 		relative
complements 		complements
in Ω {\displaystyle \Omega } 		countable
intersections 		countable
unions 		contains Ω {\displaystyle \Omega } contains ∅ {\displaystyle \varnothing } Finite
Intersection
Property

Additionally, a semiring is a Ο€-system where every complement B ∖ A {\displaystyle B\setminus A} is equal to a finite disjoint union of sets in F . {\displaystyle {\mathcal {F}}.}
A semialgebra is a semiring where every complement Ω ∖ A {\displaystyle \Omega \setminus A} is equal to a finite disjoint union of sets in F . {\displaystyle {\mathcal {F}}.}
A , B , A 1 , A 2 , … {\displaystyle A,B,A_{1},A_{2},\ldots } are arbitrary elements of F {\displaystyle {\mathcal {F}}} and it is assumed that F ≠ ∅ . {\displaystyle {\mathcal {F}}\neq \varnothing .}