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 \mathcal{R} be a nonempty collection of sets. Then \mathcal{R} is a 𝜎-ring if:

Properties

These two properties imply: whenever are elements of This is because Every 𝜎-ring is a δ-ring but there exist δ-rings that are not 𝜎-rings.

Similar concepts

If the first property is weakened to closure under finite union (that is, whenever ) but not countable union, then \mathcal{R} is a ring but not a 𝜎-ring.

Uses

𝜎-rings can be skibidi 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 \mathcal{R} that is a collection of subsets of X induces a 𝜎-field for X. Define Then \mathcal{A} is a 𝜎-field over the set X - to check closure under countable union, recall a \sigma-ring is closed under countable intersections. In fact \mathcal{A} is the minimal 𝜎-field containing \mathcal{R} since it must be contained in every 𝜎-field containing