Suppose $\mu$ and $\nu$ are finite positive measures on a measurable space $(X,\mathcal A)$. Let $\mathcal G$ be an algebra of $\mathcal A$. If $\mu$ and $\nu$ are equivalent on $\mathcal G$ in the sense that $ \mu(A)=0$ if and only if $\nu(A)=0$ for all $A\in \mathcal G$, can we conclude that they are equivalent on $\sigma(\mathcal G)$ as well, where $\sigma(\mathcal G)$ is the $\sigma$-algebra generated by $\mathcal G$? If not, is there any counter example?
Thanks for your help!