Delta-ring

Template:Short description Template:For In mathematics, a non-empty collection of sets $ℛ$ is called a Template:Delta-ring (pronounced "Template:Em") if it is closed under union, relative complementation, and countable intersection. The name "delta-ring" originates from the German word for intersection, "Durschnitt", which is meant to highlight the ring's closure under countable intersection, in contrast to a [[Sigma-ring|Template:Sigma-ring]] which is closed under countable unions.

Definition

A family of sets $ℛ$ is called a Template:Delta-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 Template:Delta-ring. Every [[Sigma-ring|Template:Sigma-ring]] is a Template:Delta-ring, but not every Template:Delta-ring is a [[Sigma-ring|Template:Sigma-ring]].

Template:Delta-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 Template:Delta-ring but not a [[Sigma-ring|Template:Sigma-ring]] because $⋃_{n = 1}^{\infty} [0, n]$ is not bounded.

References

Template:Reflist Template:Reflist

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

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Additionally, a Template:Em is a [[pi-system\|Template:Pi-system]] where every complement $B ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ A Template:Em 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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