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 $\mathcal P(\mathbb N) \subseteq \mathcal G$.



Approximating an arbitrary $\sigma$-algebra by simpler $\sigma$-algebras
It turns out that it is impossible to represent any $\sigma$-complete Boolean algebra as a countable union of Boolean subalgebras.


For example, in the paper, Boolean algebras as unions of chains of subalgebras by Sabine Koppelberg, the following result is proven.



> 
> $\mathbf{Theorem}$ Suppose $B$ is a Boolean algebra such that whenever
>  $R,S\subseteq B$ are countable subsets and $r\wedge s=0$ for $r\in R,s\in S$ then there is
>  some $b\in B$ where $r\leq b$ for each $r\in R$ and $s\leq b'$ for
>  each $s\in S$. Then whenever $B\_{n}$ is an increasing sequence of
>  proper subalgebras of $B$, then $B\neq\bigcup\_{n}B\_{n}$.
> 
> 
> 


Therefore to represent a $\sigma$-algebra as a sequence of simpler Boolean algebras, one would need a directed system of Boolean subalgebras or at least an uncountable sequence of partitions.


I also must mention that approximating Boolean algebras by partitions has been a fruitful endeavor for me. My dissertation (Boolean Partition Algebras 2013) and some of my further research has dealt with representing Boolean algebras as directed unions of subalgebras which are generated by partitions. Define a Boolean partition algebra to be a pair $(B,F)$ such that $B$ is a Boolean algebra and $F$ is a filter on the meet-semilattice of partitions of $B$ such that $B=\{0\}\cup\bigcup F$. If $p\in F$, then define $p^{\*}=\{\bigvee R|R\subseteq p,\bigvee R\,\,\textrm{exists}\}$. It is not to hard to show that each $p^{\*}$ is a Boolean subalgebra of $B$. Furthermore, it is easy to show that if $(B,F)$ is a Boolean partition algebra, then $B=\bigcup\_{p\in F}p^{\*}$. We say that a partition $p$ is subcomplete if the Boolean algebra $p^{\*}$ is a complete Boolean algebra. I found Boolean partition algebras to be useful since the category of Boolean partition algebras is equivalent to other categories such as the category of all complete non-Archimedean uniform frames and it is contravariantly equivalent to the category of pro-sets with surjective transitional mappings (pro-sets are inverse systems of sets). Furthermore, ideas such as taking quotient Boolean algebras and taking subalgebras translate nicely into taking sub-inverse systems and taking quotient inverse systems. 


Furthermore, the notion of a $\sigma$-ideal, $\sigma$-subalgebra, and $\sigma$-homomorphism between $\sigma$-complete Boolean algebras can all be represented as special ideals, subalgebras, and homomorphisms between Boolean partition algebras.


