Delta-ring

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

\( A\cup B\in {\mathcal {R}} \) for all A\* {\displaystyle A,B\in {\mathcal {R}},} \)
\( {\displaystyle A-B\in {\mathcal {R}}} \) for all \( {\displaystyle A,B\in {\mathcal {R}},} \)
\( {\displaystyle \bigcap _{n=1}^{\infty }A_{n}\in {\mathcal {R}}} \) 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.
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.