Let $\Omega$ be a set and $F$ be a field of subsets of $\Omega$. Let $\mu$ be a $\sigma$-finite measure on $F$. Then show that, $\forall A \in \sigma(F) , \forall \epsilon > 0, \exists B \in F_{\sigma}$ such that $\mu(B \text{ \ }A) < \epsilon$
I want to do it for two cases : (1) $\mu$ is finite measure and (2) $\mu$ is not finite but $\sigma$-finite.
I believe once case (1) is done we could argue like the proof of Caratheodory Extension Theorem, but unable to give a proof.
Thanks in advance for help!