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 flipping two coins.  Let $A$ be the event that the first coin is heads, and $B$ the event that the second coin is heads.   $A$ and $B$ together generate the $\sigma$-algebra of all possible events.  Suppose we know that $P(A) = P(B) = 1/2$ (i.e. each coin is unbiased).  This is not enough information to determine whether the two coins are independent, so $P$ is not completely determined.
This is the same counterexample that I gave in this answer to another question.  In its notation, $P$ and $Q$ agree on the events in $\mathcal{L}$, and $\sigma(\mathcal{L}) = \mathcal{F}$, but $P \ne Q$.
