I need help understanding a certain part of the following proof.
Theorem: (Monotone Class Theorem)
Let be a class of subsets of , closed under finite intersections and containing . Let be the smallest class containing which is closed under increasing limits and by difference. Then .
The intersection of classes of sets closed under increasing limits and differences is a class of that type. So by taking the intersection of all such classes, there always exists a smallest class containing which is closed under increasing limits and by differences. For each set , denote
to be the collection of sets such that and .
Clearly is closed under increasing limits and by difference.
Let ; for each one has
and , so .
HERE THE PROBLEM STARTS
Now let . For each we have , and because of the preceeding
, hence , whence , hence .
The rest of the proof is just concluding that is a -algebra.
Back to the problem part. Why must ?
It's easy to see when , because is closed under finite intersections.