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?

Thank you!