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