# Sigma-ring

Short description: Ring closed under countable unions

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

1. Closed under countable unions: $\displaystyle{ \bigcup_{n=1}^{\infty} A_{n} \in \mathcal{R} }$ if $\displaystyle{ A_{n} \in \mathcal{R} }$ for all $\displaystyle{ n \in \N }$
2. Closed under relative complementation: $\displaystyle{ A \setminus B \in \mathcal{R} }$ if $\displaystyle{ A, B \in \mathcal{R} }$

## Properties

These two properties imply: $\displaystyle{ \bigcap_{n=1}^{\infty} A_n \in \mathcal{R} }$ whenever $\displaystyle{ A_1, A_2, \ldots }$ are elements of $\displaystyle{ \mathcal{R}. }$

This is because $\displaystyle{ \bigcap_{n=1}^\infty A_n = A_1 \setminus \bigcup_{n=2}^{\infty}\left(A_1 \setminus A_n\right). }$

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, $\displaystyle{ A \cup B \in \mathcal{R} }$ whenever $\displaystyle{ A, B \in \mathcal{R} }$) but not countable union, then $\displaystyle{ \mathcal{R} }$ 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 $\displaystyle{ \mathcal{R} }$ that is a collection of subsets of $\displaystyle{ X }$ induces a 𝜎-field for $\displaystyle{ X. }$ Define $\displaystyle{ \mathcal{A} = \{ E \subseteq X : E \in \mathcal{R} \ \text{or} \ E^c \in \mathcal{R} \}. }$ Then $\displaystyle{ \mathcal{A} }$ is a 𝜎-field over the set $\displaystyle{ X }$ - to check closure under countable union, recall a $\displaystyle{ \sigma }$-ring is closed under countable intersections. In fact $\displaystyle{ \mathcal{A} }$ is the minimal 𝜎-field containing $\displaystyle{ \mathcal{R} }$ since it must be contained in every 𝜎-field containing $\displaystyle{ \mathcal{R}. }$

## See also

• δ-ring – Ring closed under countable intersections
• Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
• 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions
• Measurable function – Function for which the preimage of a measurable set is measurable
• π-system – Family of sets closed under intersection
• Ring of sets – Family closed under unions and relative complements
• Sample space – Set of all possible outcomes or results of a statistical trial or experiment
• 𝜎 additivity – Mapping function
• Σ-algebra – Algebraic structure of set algebra
• 𝜎-ideal – Family closed under subsets and countable unions

## References

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