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| In [[mathematics]], a nonempty collection of [[Set (mathematics)|sets]] <math>\mathcal{R}</math> is called a '''δ-ring''' (pronounced ''delta-ring'') if it is [[closure (mathematics)|closed]] under [[union (set theory)|union]], [[Complement (set theory)|relative complementation]], and countable [[Intersection (set theory)|intersection]]:
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| #<math>A \cup B \in \mathcal{R}</math> if <math>A, B \in \mathcal{R}</math>
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| #<math>A - B \in \mathcal{R}</math> if <math>A, B \in \mathcal{R}</math>
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| #<math>\bigcap_{n=1}^{\infty} A_{n} \in \mathcal{R}</math> if <math>A_{n} \in \mathcal{R}</math> for all <math>n \in \mathbb{N}</math>
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| If only the first two properties are satisfied, then <math>\mathcal{R}</math> is a [[Ring of sets|ring]] but not a δ-ring. Every [[Sigma-ring|σ-ring]] is a δ-ring, but not every δ-ring is a σ-ring.
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| δ-rings can be used instead of [[Sigma-algebra|σ-fields]] in the development of [[Measure (mathematics)|measure]] theory if one does not wish to allow sets of infinite measure.
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| == See also ==
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| *[[Ring of sets]]
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| *[[Sigma-algebra|Sigma field]]
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| *[[Sigma ring]]
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| == References ==
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| * Cortzen, Allan. "Delta-Ring." From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Delta-Ring.html
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| {{Mathanalysis-stub}}
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| [[Category:Measure theory]]
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| [[Category:Set families]]
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Latest revision as of 18:17, 12 January 2015
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