Delta-ring

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:

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

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 $𝒦 = {S \subseteq ℝ : S is bounded}$ is a $δ$ -ring but not a 𝜎-ring because $⋃_{n = 1}^{\infty} [0, n]$ is not bounded.

References

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

Families $ℱ$ of sets over $Ω$ v t e
Is necessarily true of $ℱ :$ or, is $ℱ$ closed under:	Directed by $\supseteq$	$A \cap B$	$A \cup B$	$B ∖ A$	$Ω ∖ A$	$A_{1} \cap A_{2} \cap \dots$	$A_{1} \cup A_{2} \cup \dots$	$Ω \in ℱ$	$\emptyset \in ℱ$	F.I.P.
$π$ -system			Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:
Semiring			Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:		Never
Semialgebra (Semifield)			Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:		Never
Monotone class	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	only if $A_{i} ↘$	only if $A_{i} ↗$	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:
𝜆-system (Dynkin System)		Error creating thumbnail:	Error creating thumbnail:	only if $A \subseteq B$		Error creating thumbnail:	only if $A_{i} ↗$ or they are disjoint			Never
Ring (Order theory)				Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:
Ring (Measure theory)					Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:		Never
δ-Ring					Error creating thumbnail:		Error creating thumbnail:	Error creating thumbnail:		Never
𝜎-Ring					Error creating thumbnail:			Error creating thumbnail:		Never
Algebra (Field)						Error creating thumbnail:	Error creating thumbnail:			Never
𝜎-Algebra (𝜎-Field)										Never
Filter				Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:			Error creating thumbnail:	Error creating thumbnail:
Proper filter				Never	Never	Error creating thumbnail:			Never
Prefilter (Filter base)		Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:
Filter subbase	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:
Open Topology				Error creating thumbnail:	Error creating thumbnail:	Error creating thumbnail:	File:Green check.svg (even arbitrary $\cup$ )			Never
Closed Topology				Error creating thumbnail:	Error creating thumbnail:	File:Green check.svg (even arbitrary $\cap$ )	Error creating thumbnail:			Never
Is necessarily true of $ℱ :$ or, is $ℱ$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in $Ω$	countable intersections	countable unions	contains $Ω$	contains $\emptyset$	Finite Intersection Property
Additionally, a *semiring* is a $π$ -system where every complement $B ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ A *semialgebra* is a semiring where every complement $Ω ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ $A, B, A_{1}, A_{2}, \dots$ are arbitrary elements of $ℱ$ and it is assumed that $ℱ \neq \emptyset .$

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Delta-ring

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Delta-ring

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