TITLE: Finite measure on the power set
QUESTION [6 upvotes]: Let $X$ be an uncountable set, and let $\Omega$ be the power set of $X$, viewed as a $\sigma$-algebra. Does there exist a positive $\sigma$-additive measure of finite total mass on $(X, \Omega)$ such that each point of $X$ has measure zero?

REPLY [3 votes]: Just to complement Andres's excellent answer with another reference, you can find a nice summary of the status of this question, as well as further references, in chapter 1.12(x) of Bogachev's monograph "Measure Theory I". 
The (very) short summary is that in all concrete cases the answer is no.