Consider a set X, define by .

Let , the set of all subsets in X.

Define

Prove that is a measure.

Proof.

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.

Thank you!