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:

1. ${\displaystyle A\cup B\in {\mathcal {R}}}$ for all ${\displaystyle A,B\in {\mathcal {R}},}$
2. ${\displaystyle A-B\in {\mathcal {R}}}$ for all ${\displaystyle A,B\in {\mathcal {R}},}$
3. ${\displaystyle \bigcap _{n=1}^{\infty }A_{n}\in {\mathcal {R}}}$ 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.

Example

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

See also

• Ring of sets
• Sigma field
• Sigma ring

References

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