In mathematics, a non-empty collection of sets R {\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 R {\displaystyle {\mathcal {R}}} is called a δ-ring if it has all of the following properties:
Closed under finite unions: A ∪ B ∈ R {\displaystyle A\cup B\in {\mathcal {R}}} for all A , B ∈ R , {\displaystyle A,B\in {\mathcal {R}},}
Closed under relative complementation: A − B ∈ R {\displaystyle A-B\in {\mathcal {R}}} for all A , B ∈ R , {\displaystyle A,B\in {\mathcal {R}},} and Closed under countable intersections: ⋂ n = 1 ∞ A n ∈ R {\displaystyle \bigcap _{n=1}^{\infty }A_{n}\in {\mathcal {R}}} if A n ∈ R {\displaystyle A_{n}\in {\mathcal {R}}} for all n ∈ N . {\displaystyle n\in \mathbb {N} .}
If only the first two properties are satisfied, then R {\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 K = { S ⊆ R : S is bounded } {\displaystyle {\mathcal {K}}=\{S\subseteq \mathbb {R} :S{\text{ is bounded}}\}} is a δ-ring but not a 𝜎-ring because ⋃ n = 1 ∞ [ 0 , n ] {\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
References
Cortzen, Allan. "Delta-Ring." From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Delta-Ring.html