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:
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} }
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}}.}
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