Delta-ring

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

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

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

If only the first two properties are satisfied, then ${\displaystyle {\mathcal {R}}}$ 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 ${\displaystyle {\mathcal {K}}=\{S\subseteq \mathbb {R} :S{\text{ is bounded}}\}}$ is a δ-ring but not a 𝜎-ring because ${\textstyle \bigcup _{n=1}^{\infty }[0,n]}$ 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
• Monotone class – Measure theory and probability theoremPages displaying short descriptions of redirect targets
• 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 – Family of sets closed under countable unions