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