with some examples.
Example 1.20
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 class of 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.
For property (1)
