Sigma-ring

In mathematics, a nonempty collection of sets is called a sigma-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation. Let be a nonempty collection of sets. Then is a sigma-ring if: Closed under countable unions: if for all Closed under relative complementation: if These two properties imply: whenever are elements of This is because Every sigma-ring is a δ-ring but there exist δ-rings that are not sigma-rings. 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 sigma-ring. sigma-rings can be used instead of sigma-fields (sigma-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every sigma-field is also a sigma-ring, but a sigma-ring need not be a sigma-field. A sigma-ring that is a collection of subsets of induces a sigma-field for Define Then is a sigma-field over the set - to check closure under countable union, recall a -ring is closed under countable intersections.