My Professor stated the following statement: Let $(\Sigma,\nu,\mathcal{F})$ be a measure space, $\mathcal{F}$ a sigma-algebra. Let $\mathcal{F}_n$ be a filtration, and let $A$ be a measurable set. Then $\forall\varepsilon>0 \exists n$ and a set $A_n$ such that $A_n\in\mathcal{F}_n$ and $\nu(A\triangle A_n)<\varepsilon$.
I'm looking for a source for this claim (or a generalization) of it, mainly to be able to state it properly and write a citation, but if there's a source with proof (for personal interest mainly) that will be great!
I appreciate any help :)
Edit: In the comments I was referred to a great post here Approximating a σ-algebra by a generating algebra . In it I'm mostly interested in a comment made by the user shalop, on the use of the martingale convergence theorem. Although it says it can be done "easily enough", I didn't succeed. For easy access, the theorem discussed in the post is
Theorem. Let (X,B,μ) a finite measure space, where μ is a positive measure. Let A⊂B an algebra generating B.
Then for all B∈B and ε>0, we can find A∈A such that μ(AΔB)=μ(A∪B)−μ(A∩B)<ε.
and the answer is:
Let F=σ(Ei:i∈N) be countably generated. For each n let Fn:=σ(Ei:i≤n) so that Fn form a filtration. Now take a set A∈F and consider the martingale Xn:=E[1A|Fn]. This converges a.s. to 1A, from which one may conclude easily enough (just to clarify, this would only prove that F can be approximated arbitrary closely by elements of the algebra which is generated by the Ei)...
[I would have asked there myself but can't comment]
