Is measure similar with continuous functions?

Question

suppose that $X$ is a non void set and $S$ is non empty subsets of $P(X)$. Also suppose that $M=M(S)$ [$M(S)$ is a sigma algebra generated by $S$] and $\phi_1 ,\phi_2$ are measures on $M$ such that $\phi_1=\phi_2$ on $S$.is it true that $\phi_1=\phi_2$ on $M$?

Answer 1

Let $X=\{0,1\}$ and $S=\{ \{ 0 \} \}.$ Then, $M(S) = \left\{ \emptyset, \{0\}, \{1\}, \{0,1\} \right\}.$ We must then have $\phi_1(\emptyset) = 0 = \phi(\emptyset)$ and $\phi_1(\{0\}) = \phi_2(\{0\}),$ but nothing hinders us from having $\phi_1(\{1\}) \neq \phi_2(\{1\})$ and then also $\phi_1(\{0,1\}) \neq \phi_2(\{0,1\}).$