Let be a set.
Define , where means cardinality.
Then is a -algebra. Define a measure on this measurable space by by
Show that is a measure, i.e. show that and that is -additive.
That I guess comes easily since is countable. If , pairwise disjoint:
Assume all are countable, then the countable union is countable, so .
Also, , so we have equality in this case.
If we assume that two sets are uncountable, then are countable and since we have is countable.
Thus, , but
I know I am wrong, please point out where.