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