Let be any set. On one extreme, the class consisting of two sets, and , is closed under any operation so that is a field, a ‐field, and a monotone class. On the other extreme, the class of all subsets of is also closed under any operations, finite or not, and hence is a field, a ‐field, and a monotone class. These two classes of subsets of form the smallest and the largest fields (‐field, monotone class).
For any event , it is easy to check that the class , consisting of the four events , is closed under any operations: unions, intersections, and complements of members of are again members of . This class is an example of a field (‐field, monotone class) that contains the events and , and it is the smallest such field (‐field, monotone class).
On the other hand, the class , consisting of events , is a monotone class, but neither a field nor ‐field. If and are two events, then the smallest field containing and must contain also the sets , the intersections , as well as their unions and . The closure property implies that unions such as , must also belong to .
We are now ready to present the final step.
Theorem 1.4.3 For any nonempty classof subsets of, there exists a unique smallest field (‐field, monotone class) containing all sets in. It is called the field (‐field, monotone class) generated by.
Proof We will prove the assertion for fields. Observe first that if and are fields, then their intersection (i.e., the class of sets that belong to both and ) is also a field. For instance, if (, then because each is a field, and consequently . A similar argument holds for intersections and complements.
Note that if and contain the class , then the intersection also contains . The property extends to any intersection of fields containing (not only the intersections of two such fields).
Now, let be the intersection of all fields containing . We claim that is the minimal unique field containing . We have to show that (1) exists, (2) is a field containing , (3) is unique, and (4) is minimal.
