Show set function is a measure

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

Re: Show set function is a measure

You have to assume that is uncountable (if it's not the case is not well defined). In this case, you cannot have two disjoint set whose complement is countable: if and are such sets, then is countable but it's the complement in of the empty set since and are disjoint.

Re: Show set function is a measure

So if there cant exist two disjoint sets whose complements are countable, then the sum of measures can be at most 1 also, and equal to one when just one is uncountable ?

Re: Show set function is a measure

Re: Show set function is a measure