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Sigma-ring

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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

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Let be a nonempty collection of sets. Then is a 𝜎-ring if:

  1. Closed under countable unions: if for all
  2. Closed under relative complementation: if

Properties

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These two properties imply: whenever are elements of

This is because

Every 𝜎-ring is a δ-ring but there exist δ-rings that are not 𝜎-rings.

Similar concepts

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If the first property is weakened to closure under finite union (that is, whenever ) but not countable union, then is a ring but not a 𝜎-ring.

Uses

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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 that is a collection of subsets of induces a 𝜎-field for Define Then is a 𝜎-field over the set - to check closure under countable union, recall a -ring is closed under countable intersections. In fact is the minimal 𝜎-field containing since it must be contained in every 𝜎-field containing

See also

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  • 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 algebra
  • 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions
  • Measurable function – Function for which the preimage of a measurable set is measurable
  • Monotone class – theorem
  • 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

References

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  • Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses 𝜎-rings in development of Lebesgue theory.