In mathematics, a nonempty collection of sets is called a -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 -ring if: Closed under countable unions: if for all Closed under relative complementation: if
Properties
These two properties imply:
whenever are elements of
This is because
Every -ring is a δ-ring but there exist δ-rings that are not -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 -ring.
Uses
-rings can be used instead of -fields (-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every -field is also a -ring, but a -ring need not be a -field.
A -ring that is a collection of subsets of induces a -field for Define Then is a -field over the set - to check closure under countable union, recall a -ring is closed under countable intersections. In fact is the minimal -field containing since it must be contained in every -field containing
See also
References
Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses -rings in development of Lebesgue theory.