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

be a nonempty collection of sets is a -ring if:

1. Closed under countable unions

2. Closed under relative complementation

Properties

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

Similar concepts

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 induces a -field is a is the minimal since it must be contained in every

See also

References

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