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Delta-ring

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In mathematics, a non-empty collection of sets is called a δ-ring (pronounced "delta-ring") if it is closed under union, relative complementation, and countable intersection. The name "delta-ring" originates from the German word for intersection, "Durschnitt", which is meant to highlight the ring's closure under countable intersection, in contrast to a 𝜎-ring which is closed under countable unions.

Definition

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A family of sets is called a δ-ring if it has all of the following properties:

  1. Closed under finite unions: for all
  2. Closed under relative complementation: for all and
  3. Closed under countable intersections: if for all

If only the first two properties are satisfied, then is a ring of sets but not a δ-ring. Every 𝜎-ring is a δ-ring, but not every δ-ring is a 𝜎-ring.

δ-rings can be used instead of σ-algebras in the development of measure theory if one does not wish to allow sets of infinite measure.

Examples

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The family is a δ-ring but not a 𝜎-ring because is not bounded.

See also

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  • Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
  • 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions
  • Monotone class – theorem
  • π-system – Family of sets closed under intersection
  • Ring of sets – Family closed under unions and relative complements
  • σ-algebra – Algebraic structure of set algebra
  • 𝜎-ideal – Family closed under subsets and countable unions
  • 𝜎-ring – Family of sets closed under countable unions

References

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