Let $(X,\mathcal{A},p)$ be a probability space. Let $\{\mathcal{B}_i\}_{i\in I}$ be a collection of sub-sigma-algebras of $\mathcal{A}$. Now let $$ \mathcal{B} = \bigvee_{i\in I} \mathcal{B}_i $$ be the sigma-algebra generated by all the $\mathcal{B}_i$ (which is not their union in general).
Let now $f$ be a function which is measurable (let's say even integrable) for $\mathcal{B}$. Can we find a sequence $\{f_n:X\to\Bbb{R}\}_{n\in\Bbb{N}}$ of functions such that
- each $f_n$ is measurable for some $\mathcal{B}_i$;
- $f_n\to f$ in $L^1(X,\mathcal{A},p)$?
If the answer is negative, can we improve on it by setting conditions on the $\mathcal{B}_i$ (for example, that they form a countable increasing sequence)?
The answer is negative: just enumerate all intervals with rational endpoints and take the sigma-algebras generated by them.
But for increasing sigma-algebras this is true: the union of sigma-algebras is an algebra, and we can approximate with its elements in measure any set of the generated sigma-algebra. From sets proceed to simple functions and then to arbitrary measurable ones.