Delta-ring

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In mathematics, a non-empty collection of sets $\mathcal{R}$ is called a {{delta}}-ring (pronounced "{{em|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 {{sigma}}-ring which is closed under countable unions.

Definition

A family of sets $\mathcal{R}$ is called a {{delta}}-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 $n \in \N.$

If only the first two properties are satisfied, then $\mathcal{R}$ is a ring of sets but not a {{delta}}-ring. Every {{sigma}}-ring is a {{delta}}-ring, but not every {{delta}}-ring is a {{sigma}}-ring.

{{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 $\mathcal{K} = \{ S \subseteq \mathbb{R} : S \text{ is bounded} \}$ is a {{delta}}-ring but not a {{sigma}}-ring because $\bigcup_{n=1}^{\infty} [0, n]$ 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

Category:Measure theory

Category:Families of sets

Delta-ring

Definition

Examples

See also

References