Let $\rho_1$, $\rho_2$ be two measures(not necessarily nonnegative) on $(\Omega,\mathcal{F})$, where $\Omega$ is a set, and $\mathcal{F}$ is a $\sigma$-field in $\Omega$. Let $\mathcal{F}_0$ be a field in $\Omega$ and assume that $\mathcal{F}$ is the $\sigma$-field generated by $\mathcal{F}_0$. Suppose that $\rho_1$, $\rho_2$ are $\sigma$-finite over $\mathcal{F}_0$. (Namely, we can find a countable partition $\cup\Omega_j$ of $\Omega$, where each $\Omega_j\in\mathcal{F}_0$ and $\rho_1(\Omega_j)<\infty$, $\rho_2(\Omega_j)<\infty$.)

If $\rho_1(A)\le \rho_2(A)$ for any $A\in \mathcal{F}_0$, then do we have $\rho_1(A)\le \rho_2(A)$ for any $A\in \mathcal{F}$?