Delta-ring

Short description: Ring closed under countable intersections

For $p$-derivations used in commutative algebra to define prismatic cohomology, see P-derivation.

In mathematics, a non-empty collection of sets ${ℛ}$ 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, "Durchschnitt", 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 ${ℛ}$ is called a δ-ring if it has all of the following properties:

1. Closed under finite unions: $A\cup B\in {ℛ}$ for all $A,B\in {ℛ},$
2. Closed under relative complementation: $A-B\in {ℛ}$ for all $A,B\in {ℛ},$ and
3. Closed under countable intersections: $\bigcap _{n=1}^{\mathrm{\infty }}{A}_n\in {ℛ}$ if $A_n\in {ℛ}$ for all $n\in \mathbb{\mathbb{N}}.$

If only the first two properties are satisfied, then ${ℛ}$ 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 ${𝒦}=\{S\subseteq \mathbb{ℝ}:S\text{ is bounded}\}$ is a δ-ring but not a 𝜎-ring because ${\bigcup }_{n=1}^{\mathrm{\infty }}[0,n]$ 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
• 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