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.

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}}}

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.

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.

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}}.}

• 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 – Kind of mathematical function
• Monotone class – Measure theory and probability theoremPages displaying short descriptions of redirect targets
• 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

• 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 e
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.
π-system
Semiring										Never
Semialgebra (semifield)										Never
Monotone class						only if A i ↘ {\displaystyle A_{i}\searrow }	only if A i ↗ {\displaystyle A_{i}\nearrow }
𝜆-system (Dynkin system)				only if A ⊆ B {\displaystyle A\subseteq B}			only if A i ↗ {\displaystyle A_{i}\nearrow } 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
Filter
Proper filter				Never	Never				Never
Prefilter (filter base)
Filter subbase
Open topology							(even arbitrary ∪ {\displaystyle \cup } )			Never
Closed topology						(even arbitrary ∩ {\displaystyle \cap } )				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 π-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 .}