I'm wondering if there's an example of a measure $\mu$ on a $\sigma$-algebra $\mathcal{F}$ that is $\sigma$-finite on $\mathcal{F}$ but not on $\mathcal{F}_0$, the field that generates $\mathcal{F} := \sigma\langle\mathcal{F}_0\rangle$?
Also, a related question: An example when $\mu$ is $\sigma$-finite on $\mathcal{F}_0 \cup \mathcal{N}$ where $\mathcal{N} = \{F \in \mathcal{F}: \mu[F] = 0\}$, but not on $\mathcal{F}_0$? And an example where $\mu$ is $\sigma$-finite on $\mathcal{F}$ but not on $\mathcal{F}_0 \cup \mathcal{N}$?