Delta-ring

From Quire, the open reference·Ring closed under countable intersections·1 min read

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#edit

A family of sets ${\displaystyle {\mathcal {R$ is called a δ-ring if it has all of the following properties:

Closed under finite unions: ${\displaystyle A\cup B\in {\mathcal {R$ for all ${\displaystyle A,B\in {\mathcal {R,}$
Closed under relative complementation: ${\displaystyle A-B\in {\mathcal {R$ for all ${\displaystyle A,B\in {\mathcal {R,}$ and
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#edit

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.

References#edit

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

Families ${\displaystyle {\mathcal {F$ of sets over ${\displaystyle \Omega }$
Is necessarily true of ${\displaystyle {\mathcal {F\colon }$ or, is ${\displaystyle {\mathcal {F$ closed under:	Directed by ${\displaystyle \,\supseteq }$	${\displaystyle A\cap B}$	${\displaystyle A\cup B}$	${\displaystyle B\setminus A}$	${\displaystyle \Omega \setminus A}$	${\displaystyle A_{1}\cap A_{2}\cap \cdots }$	${\displaystyle A_{1}\cup A_{2}\cup \cdots }$	${\displaystyle \Omega \in {\mathcal {F$	${\displaystyle \varnothing \in {\mathcal {F$	F.I.P.
π-system
Semiring										Never
Semialgebra (semifield)										Never
Monotone class						only if ${\displaystyle A_{i}\searrow }$	only if ${\displaystyle A_{i}\nearrow }$
𝜆-system (Dynkin system)				only if ${\displaystyle A\subseteq B}$			only if ${\displaystyle A_{i}\nearrow }$ or they are disjoint			Never
Ring (order theory)
Ring (measure theory)										Never
δ-ring										Never
𝜎-ring										Never
Algebra (field)										Never
𝜎-algebra (𝜎-field)										Never
Filter
Proper filter				Never	Never				Never
Prefilter (filter base)
Filter subbase
Open topology							(even arbitrary ${\displaystyle \cup }$ )			Never
Closed topology						(even arbitrary ${\displaystyle \cap }$ )				Never
Is necessarily true of ${\displaystyle {\mathcal {F\colon }$ or, is ${\displaystyle {\mathcal {F$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in ${\displaystyle \Omega }$	countable intersections	countable unions	contains ${\displaystyle \Omega }$	contains ${\displaystyle \varnothing }$	Finite intersection property
Additionally, a *semiring* is a π-system where every complement ${\displaystyle B\setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F.}$ A *semialgebra* is a semiring where every complement ${\displaystyle \Omega \setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F.}$ ${\displaystyle A,B,A_{1},A_{2},\ldots }$ are arbitrary elements of ${\displaystyle {\mathcal {F$ and it is assumed that ${\displaystyle {\mathcal {F\neq \varnothing .}$

This article is mirrored from “Delta-ring” on English Wikipedia, revision 1365899930, written by Wikipedia's contributors and licensed CC BY-SA 4.0. Editing it here creates a Quire copy with its own history.

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Delta-ring

Definition#edit

Examples#edit

See also#edit

References#edit