Sigma-ring
# 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.

## Formal definition

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

* Closed under countable unions: $\bigcup _{n=1}^{\infty }A_{n}\in {\mathcal {R}}$ if $A_{n}\in {\mathcal {R}}$ for all $n\in \mathbb {N}$
* Closed under relative complementation: $A\setminus B\in {\mathcal {R}}$ if $A,B\in {\mathcal {R}}$

## Properties

These two properties imply:

$\bigcap _{n=1}^{\infty }A_{n}\in {\mathcal {R}}$

whenever $A_{1},A_{2},\ldots$ are elements of ${\mathcal {R}}.$

This is because

$\bigcap _{n=1}^{\infty }A_{n}=A_{1}\setminus \bigcup _{n=2}^{\infty }\left(A_{1}\setminus A_{n}\right).$

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, $A\cup B\in {\mathcal {R}}$ whenever $A,B\in {\mathcal {R}}$) but not countable union, then ${\mathcal {R}}$ 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 ${\mathcal {R}}$ that is a collection of subsets of $X$ induces a 𝜎-field for $X.$ Define ${\mathcal {A}}=\{E\subseteq X:E\in {\mathcal {R}}\ {\text{or}}\ E^{c}\in {\mathcal {R}}\}.$ 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 ${\mathcal {R}}.$

