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

1. Closed under countable unions: $\bigcup _{n=1}^{\mathrm{\infty }}{A}_n\in {ℛ}$ if $A_n\in {ℛ}$ for all $n\in \mathbb{\mathbb{N}}$
2. Closed under relative complementation: $A\setminus B\in {ℛ}$ if $A,B\in {ℛ}$

Properties

These two properties imply: $$\bigcap _{n=1}^{\mathrm{\infty }}{A}_n\in {ℛ}$$ whenever $A_1,A_2,\dots$ are elements of ${ℛ}.$

This is because $$\bigcap _{n=1}^{\mathrm{\infty }}{A}_n=A_1\setminus \bigcup _{n=2}^{\mathrm{\infty }}(A_1\setminus A_n).$$

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, $A\cup B\in {ℛ}$ whenever $A,B\in {ℛ}$) but not countable union, then ${ℛ}$ 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 ${ℛ}$ that is a collection of subsets of $X$ induces a 𝜎-field for $X.$ Define ${𝒜}=\{E\subseteq X:E\in {ℛ}\text{or}{E}^c\in {ℛ}\}.$ Then ${𝒜}$ 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 ${𝒜}$ is the minimal 𝜎-field containing ${ℛ}$ since it must be contained in every 𝜎-field containing ${ℛ}.$

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.