Sigma-ring (original) (raw)

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Family of sets 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.

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

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

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

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).} $\bigcap _{n=1}^{\infty }A_{n}=A_{1}\setminus \bigcup _{n=2}^{\infty }\left(A_{1}\setminus A_{n}\right).$

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

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}}} ${\mathcal {F}}$ of sets over Ω {\displaystyle \Omega } $\Omega$ v t e
Is necessarily true of F : {\displaystyle {\mathcal {F}}\colon } ${\mathcal {F}}\colon$ or, is F {\displaystyle {\mathcal {F}}} ${\mathcal {F}}$ closed under:	Directed by ⊇ {\displaystyle \,\supseteq } $\,\supseteq$	A ∩ B {\displaystyle A\cap B} $A\cap B$	A ∪ B {\displaystyle A\cup B} $A\cup B$	B ∖ A {\displaystyle B\setminus A} $B\setminus A$	Ω ∖ A {\displaystyle \Omega \setminus A} $\Omega \setminus A$	A 1 ∩ A 2 ∩ ⋯ {\displaystyle A_{1}\cap A_{2}\cap \cdots } $A_{1}\cap A_{2}\cap \cdots$	A 1 ∪ A 2 ∪ ⋯ {\displaystyle A_{1}\cup A_{2}\cup \cdots } $A_{1}\cup A_{2}\cup \cdots$	Ω ∈ F {\displaystyle \Omega \in {\mathcal {F}}} $\Omega \in {\mathcal {F}}$	∅ ∈ F {\displaystyle \varnothing \in {\mathcal {F}}} $\varnothing \in {\mathcal {F}}$	F.I.P.
π-system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if A i ↘ {\displaystyle A_{i}\searrow } $A_{i}\searrow$	only if A i ↗ {\displaystyle A_{i}\nearrow } $A_{i}\nearrow$
𝜆-system (Dynkin System)				only if A ⊆ B {\displaystyle A\subseteq B} $A\subseteq B$			only if A i ↗ {\displaystyle A_{i}\nearrow } $A_{i}\nearrow$ orthey 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				∅ ∉ F {\displaystyle \varnothing \not \in {\mathcal {F}}} $\varnothing \not \in {\mathcal {F}}$
Prefilter (Filter base)				Never	Never				∅ ∉ F {\displaystyle \varnothing \not \in {\mathcal {F}}} $\varnothing \not \in {\mathcal {F}}$
Filter subbase				Never	Never				∅ ∉ F {\displaystyle \varnothing \not \in {\mathcal {F}}} $\varnothing \not \in {\mathcal {F}}$
Open Topology							(even arbitrary ∪ {\displaystyle \cup } $\cup$ )			Never
Closed Topology						(even arbitrary ∩ {\displaystyle \cap } $\cap$ )				Never
Is necessarily true of F : {\displaystyle {\mathcal {F}}\colon } ${\mathcal {F}}\colon$ or, is F {\displaystyle {\mathcal {F}}} ${\mathcal {F}}$ closed under:	directed downward	finiteintersections	finiteunions	relativecomplements	complementsin Ω {\displaystyle \Omega } $\Omega$	countableintersections	countableunions	contains Ω {\displaystyle \Omega } $\Omega$	contains ∅ {\displaystyle \varnothing } $\varnothing$	Finite Intersection Property
Additionally, a *semiring* is a π-system where every complement B ∖ A {\displaystyle B\setminus A} $B\setminus A$ is equal to a finite disjoint union of sets in F . {\displaystyle {\mathcal {F}}.} ${\mathcal {F}}.$ A *semialgebra* is a semiring where every complement Ω ∖ A {\displaystyle \Omega \setminus A} $\Omega \setminus A$ is equal to a finite disjoint union of sets in F . {\displaystyle {\mathcal {F}}.} ${\mathcal {F}}.$ A , B , A 1 , A 2 , … {\displaystyle A,B,A_{1},A_{2},\ldots } $A,B,A_{1},A_{2},\ldots$ are arbitrary elements of F {\displaystyle {\mathcal {F}}} ${\mathcal {F}}$ and it is assumed that F ≠ ∅ . {\displaystyle {\mathcal {F}}\neq \varnothing .} ${\mathcal {F}}\neq \varnothing .$