In mathematics, a nonempty collection of sets is called a -ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.
Let be a nonempty collection of sets. Then is a -ring if:
These two properties imply:
whenever are elements of
This is because
Every -ring is a ô-ring but there exist ô-rings that are not -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 -ring.
-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