Let $(\Omega,\mathscr F)$ be a measurable space with two probability measures $\mu, \nu: \mathscr F\to[0,1]$ defined over it. Suppose that $\mathscr C\subset\mathscr F$ is some class of sets and $$ (\mu-\nu)|_\mathscr C = 0. $$ Which necessary and sufficient conditions are known on $\mathscr C$ in order to assure that $\mu = \nu$?
For example, it is sufficient for $\mathscr C$ to be a ring of sets generating $\mathscr F$, as it follows from Caratheodory's extension theorem. However, is that condition also necessary?
In particular, if $\sigma(\mathscr C) = \mathscr F$ can it be the case that $\mu\neq \nu$?