Delta-ring

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.

Definition

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

1. Closed under finite unions: ${\displaystyle A\cup B\in {\mathcal {R}}}$ for all ${\displaystyle A,B\in {\mathcal {R}},}$
2. Closed under relative complementation: ${\displaystyle A-B\in {\mathcal {R}}}$ for all ${\displaystyle A,B\in {\mathcal {R}},}$ and
3. 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 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-fields in the development of measure theory if one does not wish to allow sets of infinite measure.

Examples

The family ${\displaystyle {\mathcal {K}}=\{S\subset \mathbb {R} :S{\text{ is bounded}}\}}$ is a δ-ring but not a σ-ring because ${\displaystyle \cup _{n=1}^{\infty }[0,n]}$ is not bounded.

See also

• Algebra of sets
• Field of sets
• λ-system (Dynkin system)
• π-system
• Ring of sets
• σ-algebra
• σ-ring

References

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