Let X be an uncountable set, let
Consider the -algebra
Define by and
Prove that is a measure.
Proof so far.
Let , claim:
Case 1) If is countable for all n, then is countable and since
Case 2) If is countable. Since and the latter is countable, I then have
But then I also have
So the two ain't equal... What did I do wrong?
Case 3) Either is countable or is countable. How should I proceed with this one?