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

A family of sets is called a Ξ΄-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.

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

Examples

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

See also

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

References

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