Delta-ring

In mathematics, a non-empty collection of sets $$\mathcal{R}$$ is called a $\delta$-ring (pronounced "") if it is closed under union, relative complementation, and countable intersection. The name "delta-ring" originates from the German word for intersection, "Durschnitt", 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 -ring if it has all of the following properties:


 * 1) Closed under finite unions: $$A \cup B \in \mathcal{R}$$ for all $$A, B \in \mathcal{R},$$
 * 2) Closed under relative complementation: $$A - B \in \mathcal{R}$$ for all $$A, B \in \mathcal{R},$$ and
 * 3) 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 -ring. Every 𝜎-ring is a -ring, but not every -ring is a 𝜎-ring.

-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 -ring but not a 𝜎-ring because $\bigcup_{n=1}^{\infty} [0, n]$ is not bounded.