Delta-ring

Short description: Ring closed under countable intersections

In mathematics, a non-empty collection of sets [math]\displaystyle{ \mathcal{R} }[/math] is called a δ-ring (pronounced "delta-ring") if it is closed under union, relative complementation, and countable intersection. The name "delta-ring" originates from the German word for intersection, "Durschnitt", which is meant to highlight the ring's closure under countable intersection, in contrast to a 𝜎-ring which is closed under countable unions.

Definition

A family of sets [math]\displaystyle{ \mathcal{R} }[/math] is called a δ-ring if it has all of the following properties:

1. Closed under finite unions: [math]\displaystyle{ A \cup B \in \mathcal{R} }[/math] for all [math]\displaystyle{ A, B \in \mathcal{R}, }[/math]
2. Closed under relative complementation: [math]\displaystyle{ A - B \in \mathcal{R} }[/math] for all [math]\displaystyle{ A, B \in \mathcal{R}, }[/math] and
3. Closed under countable intersections: [math]\displaystyle{ \bigcap_{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]

If only the first two properties are satisfied, then [math]\displaystyle{ \mathcal{R} }[/math] is a ring of sets but not a δ-ring. Every 𝜎-ring is a δ-ring, but not every δ-ring is a 𝜎-ring.

δ-rings can be used instead of σ-algebras in the development of measure theory if one does not wish to allow sets of infinite measure.

Examples

The family [math]\displaystyle{ \mathcal{K} = \{ S \subseteq \mathbb{R} : S \text{ is bounded} \} }[/math] is a δ-ring but not a 𝜎-ring because [math]\displaystyle{ \bigcup_{n=1}^{\infty} [0, n] }[/math] is not bounded.

See also

• 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
• π-system – Family of sets closed under intersection
• Ring of sets – Family closed under unions and relative complements
• Σ-algebra – Algebraic structure of set algebra
• 𝜎-ideal – Family closed under subsets and countable unions
• 𝜎-ring – Ring closed under countable unions

References

• Cortzen, Allan. "Delta-Ring." From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Delta-Ring.html