Delta-ring

In mathematics, a nonempty collection of sets ${\mathcal {R}}$ is called a δ-ring (pronounced delta-ring) if it is closed under union, relative complementation, and countable intersection:

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

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