Given a set , let be a nonempty set of subsets of . Then is a ring if, for every pair of sets in , the intersection, union, and set difference is also in . is called a -ring if is a ring and, for any countable collection of sets , the intersection is also in . A -ring is -finite if is the union of a countable collection of sets in .
Given a collection of subsets of , the -ring generated by can be defined as the intersection of all -rings containing . For example, the collection of bounded real Borel sets is a -ring. More generally, if is a Hausdorff topological space, then the collection of Borel sets with compact closure is a -ring.
Unbounded (complex) measures are defined on -rings. If is a -algebra, it is a -ring, and if it is a -ring, it is a ring.
A ring of sets in is also a ring in algebraic sense, if addition is defined as
and multiplication as