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}},} \)
$A-B\in {\mathcal {R}}$ for all $A,B\in {\mathcal {R}},$
$\bigcap _{n=1}^{\infty }A_{n}\in {\mathcal {R}}$ 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

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