Singularity is closed under sums; $\nu\ll\mu$ and $\nu\perp\mu$ force $\nu=0$

Consider the statements: I. If $\nu_1,\nu_2$ and $\mu$ are measures and $\nu_1\perp\mu$, $\nu_2\perp\mu$, then $\nu_1+\nu_2\perp\mu$. II. If $\nu$ and $\mu$ are measures such that $\nu\ll\mu$ and $\nu\perp\mu$, then $\nu$ is identically zero.