Is a measure for a sigma algebra determined by its values for a generator of the sigma algebra? If the value of a measure on any subset in a generator of a sigma algebra is known, will the measure for the sigma algebra also be uniquely determined? Thanks!

Consider the $\sigma$-algebra given by all subsets of $\{a,b,c\}$.
It is generated by $A = \{a,b\}$ and $C = \{b,c\}$.
Let $\mu(A) = \mu(C) = 1$.
It could be that $\mu(\{a\}) = \mu(\{b\}) = \mu(\{c\}) = \frac{1}{2}$.
It could also be that $\mu(\{a\}) = \mu(\{c\}) = \frac{1}{3}$ and $\mu(\{b\}) = \frac{2}{3}$.
