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
 |
