Let $\{\mathcal B\_i:i\in I\}$ be a family of $\sigma$-algebras (over the same set $\Omega$) which are totally ordered by inclusion, in the sense that for any $i,j\in I$ either $\mathcal B\_i\subset\mathcal B\_j$ or $\mathcal B\_j\subset\mathcal B\_i$. I think this can be called a filtration with time $I$.


Let $\mathcal B=\sigma\left(\bigcup\_{i\in I}\mathcal B\_i\right)$ be the $\sigma$-algebra generated by all the $\mathcal B\_i$, let $\mu$ be a probability on $\mathcal B$ and let $X\in L^2(\mathcal B,\mu)$ be a bounded random variable. 


In what senses (if any) does the family $\{\mathbb{E}[X\mid\mathcal B\_i];i\in I\}$ approximate $X$?
More precisely: 


Is it true that for every $\epsilon>0$ there is some $i\in I$ such that $\|X-\mathbb{E}[X\mid\mathcal B\_i]\|\_{L^2}<\epsilon$? ($L^2$ convergence)


Is it true that there exists a set $A\in\mathcal B$ with $\mu(A)=1$ and for every $\epsilon>0$ there is some $i\in I$ such that for each $\omega\in A$ we have $|X(\omega)-\mathbb{E}[X\mid\mathcal B\_i](\omega)|<\epsilon$? (almost sure convergence)



Does martingale convergence hold for arbitrary time?
The sigma algebra generated by $X$ is countably generated. Thus $X$ is measurable for
$$
\sigma\left(\bigcup\_{k=1}^\infty \mathcal{B}\_{i\_k}\right)
$$
for some increasing sequence $i\_1 \le i\_2 \le \dots$ in $I$. So some of your desired results follow from the usual martingale convergence theory. 


The almost sure convergence will fail, in general, for uncountable $I$. A reasonable replacement is called "essential convergence". That is what Sucheston and I used for such things in our book.


[plug: G. Edgar, L. Sucheston, [*Stopping Times and Directed Processes*](http://www.math.osu.edu/~edgar.2/books/stdp.html) (Cambridge University Press 1992)]


