σ-ring

In mathematics, a nonempty collection of sets is called a -ring (pronounced ''sigma-ring'') if it is closed under countable union and relative complementation.

Formal definition

Let $\mathcal$ be a nonempty collection of sets. Then $\mathcal$ is a -ring if:
# Closed under countable unions: $\bigcup_^ A_ \in \mathcal$ if $A_ \in \mathcal$ for all $n \in \N$
# Closed under relative complementation: $A \setminus B \in \mathcal$ if $A, B \in \mathcal$

Properties

These two properties imply:
$\bigcap_^ A_n \in \mathcal$
whenever $A_1, A_2, \ldots$ are elements of $\mathcal.$

This is because
$\bigcap_^\infty A_n = A_1 \setminus \bigcup_^\left(A_1 \setminus A_n\right).$

Every -ring is a δ-ring but there exist δ-rings that are not -rings.

Similar concepts

If the first property is weakened to closure under finite union (that is, $A \cup B \in \mathcal$ whenever $A, B \in \mathcal$) but not countable union, then $\mathcal$ is a ring but not a -ring.

Uses

-rings can be used instead of -fields (-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every -field is also a -ring, but a -ring need not be a -field.

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

See also

*
*
*
*
*
*
*
*
*
*
*
*

References

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

{{Families of sets

Measure theory
Families of sets