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 $U \subset S$, define


$$
P(U) = 
\begin{cases}
1, \quad \text{if} \quad a \in U \\
0, \quad \text{otherwise}
\end{cases}
$$


But what happens if I am not allowed to put nonzero probability on individual points ?



Existence of probability measure defined on all subsets
The existence of such a measure is equiconsistent to the existence of a [measurable cardinal](https://en.wikipedia.org/wiki/Measurable_cardinal), one of the large cardinal notions, and if ZFC is consistent, cannot be proved in ZFC. (See the notion of real-valued measurable cardinal on the Wikipedia page.)


