Hi,

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.

Thanks