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 $$ \lambda(A) = \sum\limits_{x\in A}p(x). $$