In mathematics, a nonempty collection of sets is called a **σ-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 **σ-ring** if:

- if for all
- if

## Properties

From these two properties we immediately see that

- if for all

This is simply because .

## Similar concepts

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.

## Uses

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 .

## See also

## References

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