A $\sigma$-algebra $\mathcal F$ over $\Omega$ is generated by an countable partition if there exits a countable partition $\mathcal B = \{ B_i \}$ of $\Omega$ such that $\mathcal F = \sigma(\mathcal B)$. Now let $\mathcal G$ be an arbitrary $\sigma$-algebra over $\Omega$. Is it possible to find $\sigma$-algebras $\mathcal G_n$ generated by countable partitions approximating $\mathcal G$, i.e. such that $G_1 \subseteq G_2 \subseteq G_3 \subseteq \ldots$ and $\mathcal G = \bigcup_n \mathcal G_n$?

One naive idea of me is to take some arbitrary $A_1 \in \mathcal G$, and then set $\mathcal B_1 = \{ A_1, A_1^C \}$ and $\mathcal G_1 = \sigma(\mathcal B_1)$, then take some $A_2 \in \mathcal G$ with $A_2 \cap A_1 = \emptyset$ and set $\mathcal B_2 = \{ A_1, A_2, (A_1\cup A_2)^C \}$ and $\mathcal G_2 := \sigma(\mathcal B_2)$ and so on. But this will not work, for example if $\mathcal P(\mathbb N) \subseteq \mathcal G$ and I choose $A_i := \{ i \}$, then as the "limit" $\sigma$-algebra I will get $\sigma(\{ A_i \}) = \{ M : M \subseteq \mathbb N \mbox{ or } X\setminus M \subseteq \mathbb N \}$, which has not to be the the original $\sigma$-algebra with which I started, for example if $\mathbb G = \mathcal B(\mathbb R)$ if fullfills $\mathcal P(\mathbb N) \subseteq \mathcal G$.