Now we want to move from algebras of sets closer to -algebras by defining a new structure: a “monotone class”. This is a collection so that if we have an increasing sequence of sets
with , then the “limit” of this sequence is in . Similarly, if we have a decreasing sequence of sets
with , then the “limit” of this sequence is in . In each case, we can construe this limit more simply. In the case of an increasing sequence, each term contains all the terms before it, and is thus the same as the union of the whole sequence up to that point. Thus we can define
Similarly, for a decreasing sequence we can define
Now, if is an algebra of subsets of , and is a monotone class containing , then also contains , the smallest -algebra containing .
As an aside, what do we mean by “smallest”? Well, it’s not hard to see that the intersection of a collection of algebras is itself a -algebra, just like we did way back when we showed that we had lower bounds in the lattice of ideals. So take all the -algebras containing — there is at least one, because itself is one — and define to be the intersection of all of them. This is a -algebra contained in all the others, thus “smallest”.
Similarly, there is a smallest monotone class containing ; we will assume that is this one without loss of generality, since if it contains then all the others do as well. We will show that this is actually a -algebra, and then it must contain !
Given , define to be the collection of so that . This is itself a monotone class, and it contains , and so it must contain . But it’s defined as being contained in , and so we must have . Thus as long as and .
Now take a and define to be the collection of so that . What we just showed is that , and is a monotone class. And so , and whenever both and are in .
We can repeat a similar argument to the previous two paragraphs to show that for . Start by defining as the collection of so that both and are in .
So is both a monotone class and an algebra. We need to show that it’s a -algebra by showing that it’s closed under countable unions. But if then is a monotone increasing sequence of set in , since they’re all finite unions. The countable union is the limit of this sequence, which contains by virtue of being a monotone class.
And so is a -algebra containing , and so it contains .