TITLE: Existence of probability measure defined on all subsets
QUESTION [15 upvotes]: Let S be an uncountable set. Does there exist a probability measure which is defined on all subsets of S, with P({x}) = 0 for any element x of S ?
If I remove the condition P({x}) = 0, then I can trivially get a measure defined on all subsets as follows:
Fix some a in S. For any subset U of S, define P(U ) = 1 if a is in U and 0 otherwise.
But what happens if I am not allowed to put nonzero probability on individual points ?

REPLY [9 votes]: Joel's answer is the correct one, but in some cases one only needs a finitely additive probability measure rather than a countably additive one, and in this case one can use an non-principal ultrafilter to create such a measure, which would give every set in the ultrafilter a measure of 1 and all the other sets a measure of zero.  Indeed, one important way to think about ultrafilters is as a {0,1}-valued finitely additive probability measure on a set.