A 
-ring on a set 
is a ring of sets 
closed under the countable intersections 
of sequences 
. Here a ring of sets is a nonempty set of subsets of
closed under finite unions and set differences.
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 
is the intersection of all
-rings containing 
. For example, the collection of bounded Borel sets in 
is a 
-ring. More generally, if 
is a Hausdorff topological
space, then the collection of Borel sets with compact closure is a 
-ring.
Complex-valued measures on 
-rings need not be bounded.
Every sigma-algebra is a 
-ring, and every 
-ring is a ring of sets.
A ring of sets in 
is also a ring in the algebraic sense, with addition
defined as
 |
and multiplication as
 |
