Let $(\Omega,\mathcal{A},\mu)$ be a measure space and $\mathcal B := \{A \in \mathcal A \mid \mu(A) < \infty\}$. We define a map $d_\mu:\mathcal A \times \mathcal A \to [0, \infty]$ by $d_{\mu}(A, B) := \mu(A \triangle B)$. Then $d_\mu$ becomes an extended metric when $\mathcal A$ is considered modulo the equivalence relation $\sim$ defined by $$ A \sim B \iff \mu(A \triangle B) = 0 \quad \forall A,B \in \mathcal A. $$
If there is a countable collection $(B_n) \subset \mathcal B$ such that $\mathcal A$ is generated by $(B_n)$, then $\mathcal B$ is dense in $\mathcal A$ w.r.t. the topology induced by $d_\mu$.
Are there some necessary and/or sufficient conditions such that $\mathcal B$ is dense in $\mathcal A$?