Consider a set X, define by .
Let , the set of all subsets in X.
Prove that is a measure.
Let , with all disjoint.
I claim that
Now suppose that is the largest finite subset.
Then for the biggest subset. Note that are disjoint since each is disjoint.
Then we have
It follows that
Therefore is a measure.
Is this okay? I know that idea should be right, but it is just the part about choosing the largest finite subset could be a bit verge.