This is a supplement to one of my analysis discussion sessions.
We will now prove a fascinating result in measure theory: there are no -algebras that are infinitely countable. This means that any -algebra is either finite (and is therefore just an algebra) or very 'BIG' in cardinality, in the sense that it is uncountable. The idea behind this proof is simple. We will take a -algebra on a set , and show that the collection of `smallest' non-trivial sets belonging to are in one-to-one correspondence with a power set of a certain infinite set.
Let's assume by contradiction that is an infinitely countable -algebra defined on a set . The set has to be infinite as well (otherwise any -algebra defined on it is smaller in cardinality than its power set that is finite as well.) Define a function
Namely, the function maps to the smallest set of the -algebra that contains . Note that this function is well-defined (and maps into ) exactly because of the assumption that is countable (and therefore any countable intersection remains in by the definition of a -algebra.) Since is a collection of sets, the image of under is a subset of the power set of . In fact, it turns out that provides us with a partition of .
Let's try to see why. Consider the images of two points under and assume by contradiction they have a non-trivial intersection . If then is a smaller set in the -algebra containing which is a contradiction to the definition of . Therefore and by the same argument . But then since maps a point to the smallest set in the -algebra containing it, and and thus we conclude that .
We may conclude that is a partition of . Each set in can be written as the union of such images,
Therefore, the partition cannot be finite, as otherwise would be finite as well. But now if the partition is infinite, one can form all the sets in by taking all the possible (disjoint) unions of sets in . This means that the cardinality of is equal to the cardinality of the power set of , namely
which is of course uncountable.
This is a contradiction, and therefore the -algebra cannot be infinitely countable.