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

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

If only the first two properties are satisfied, then ${\mathcal {R}}$ 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 ${\mathcal {K}}=\{S\subseteq \mathbb {R} :S{\text{ is bounded}}\}$ is a $δ$ -ring but not a 𝜎-ring because ${\textstyle \bigcup _{n=1}^{\infty }[0,n]}$ is not bounded.

References

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

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

Sample Page

Definition

Examples

See also

References