A quick question on probability measure

I have read that for a un-contable sample space, not every subset can have a probabiltiy associated to it. (If we do that the axioms of probability measure will get voilated)

Cosider this
1. S is un-countable set
2. Let A be any subset of S. Define P(A) = 1 iff A has a particular element, e0. Else P(A) = 0;

Doesn't it correctly define a probability measure? And doesn't it define a measure for 'every' subset of S?

Where am I missing the point? Thanks for any help.

Maybe the theorem you are referring to is that every non-empty subset cannot have a *positive* probability?

Thanks. I am not too sure - I thought what it said was that probability measure can't be difined on the entire power set. It can be defined only on a subset (called sigma algebra). But if 'positive' was impled there then my question is answered. Thanks