View source for Delta-ring

{{short description|Ring closed under countable intersections}}
{{for|<math>p</math>-derivations used in commutative algebra to define prismatic cohomology|P-derivation}}
In [[mathematics]], a [[Empty set|non-empty]] [[Family of sets|collection]] of [[Set (mathematics)|sets]] <math>\mathcal{R}</math> is called a '''{{delta}}-ring''' (pronounced "{{em|delta-ring}}") if it is [[Closure (mathematics)|closed]] under [[Union (set theory)|union]], [[Complement (set theory)|relative complementation]], and countable [[Intersection (set theory)|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 [[Sigma-ring|{{sigma}}-ring]] which is closed under countable unions.

== Definition ==

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

#Closed under finite unions: <math>A \cup B \in \mathcal{R}</math> for all <math>A, B \in \mathcal{R},</math>
#Closed under relative complementation: <math>A - B \in \mathcal{R}</math> for all <math>A, B \in \mathcal{R},</math> and
#Closed under countable intersections: <math>\bigcap_{n=1}^{\infty} A_n \in \mathcal{R}</math> if <math>A_n \in \mathcal{R}</math> for all <math>n \in \N.</math>

If only the first two properties are satisfied, then <math>\mathcal{R}</math> is a [[ring of sets]] but not a {{delta}}-ring. Every [[Sigma-ring|{{sigma}}-ring]] is a {{delta}}-ring, but not every {{delta}}-ring is a [[Sigma-ring|{{sigma}}-ring]].

{{delta}}-rings can be used instead of [[σ-algebra]]s in the development of [[measure theory]] if one does not wish to allow sets of infinite measure.

== Examples ==

The family <math>\mathcal{K} = \{ S \subseteq \mathbb{R} : S \text{ is bounded} \}</math> is a {{delta}}-ring but not a [[Sigma-ring|{{sigma}}-ring]] because <math display="inline">\bigcup_{n=1}^{\infty} [0, n]</math> is not bounded.

== See also ==

* {{annotated link|Field of sets}}
* {{annotated link|Dynkin system|{{lambda}}-system (Dynkin system)}}
* {{annotated link|Monotone class}}
* {{annotated link|Pi-system|{{pi}}-system}}
* {{annotated link|Ring of sets}}
* {{annotated link|σ-algebra}}
* {{annotated link|Sigma-ideal|{{sigma}}-ideal}}
* {{annotated link|Sigma-ring|{{sigma}}-ring}}

== References ==

{{reflist|group=note}}
{{reflist}}

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

{{Families of sets}}

{{Mathanalysis-stub}}

[[Category:Measure theory]]
[[Category:Families of sets]]