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

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

Properties

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

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

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

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.