Let $X = [0,1]$ and $\mathbb Q$ - the set of rational numbers. We take X' = X\cap \mathbb Q and define a measure on it such that \lambda(X'\cap (a,b)) = b-a for any $a,b\in X$.
This measure is characterized by its values on atoms since there are a countable number of elements of X'. It's easy to see that this measure is non-unique - but can you give at least one example of such a measure $\lambda$ on rational numbers in $[0,1]$?
With an example I mean a function p:X'\to [0,1] such that for any subset A\subseteq X' holds $ \lambda(A) = \sum\limits_{x\in A}p(x). $