Sigma-ring

In mathematics, a nonempty collection of sets is called a ๐œŽ-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.

Formal definition

Let be a nonempty collection of sets. Then is a ๐œŽ-ring if:

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

Properties

These two properties imply:

whenever are elements of

This is because

Every ๐œŽ-ring is a ฮด-ring but there exist ฮด-rings that are not ๐œŽ-rings.

Similar concepts

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 ๐œŽ-ring.

Uses

๐œŽ-rings can be used instead of ๐œŽ-fields (๐œŽ-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every ๐œŽ-field is also a ๐œŽ-ring, but a ๐œŽ-ring need not be a ๐œŽ-field.

A ๐œŽ-ring that is a collection of subsets of induces a ๐œŽ-field for Define Then is a ๐œŽ-field over the set - to check closure under countable union, recall a -ring is closed under countable intersections. In fact is the minimal ๐œŽ-field containing since it must be contained in every ๐œŽ-field containing

See also

  • ฮด-ring โ€“ Ring closed under countable intersections
  • Field of sets โ€“ Algebraic concept in measure theory, also referred to as an algebra of sets
  • Join (sigma algebra) โ€“ Algebric structure of set algebra
  • ๐œ†-system (Dynkin system) โ€“ Family closed under complements and countable disjoint unions
  • Measurable function โ€“ Function for which the preimage of a measurable set is measurable
  • Monotone class โ€“ theorem
  • ฯ€-system โ€“ Family of sets closed under intersection
  • Ring of sets โ€“ Family closed under unions and relative complements
  • Sample space โ€“ Set of all possible outcomes or results of a statistical trial or experiment
  • ๐œŽ additivity โ€“ Mapping function
  • ฯƒ-algebra โ€“ Algebric structure of set algebra
  • ๐œŽ-ideal โ€“ Family closed under subsets and countable unions

References

    • Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses ๐œŽ-rings in development of Lebesgue theory.
    This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.