Sigma-ring

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.

be a nonempty collection of sets.

is a 𝜎-ring if: These two properties imply:

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 (that is,

) 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

is a 𝜎-field over the set

- to check closure under countable union, recall a

-ring is closed under countable intersections.

is the minimal 𝜎-field containing

since it must be contained in every 𝜎-field containing

Additionally, a semiring is a π-system where every complement

is equal to a finite disjoint union of sets in

A semialgebra is a semiring where every complement

is equal to a finite disjoint union of sets in

are arbitrary elements of