Let $\mathcal{F}_n$ be a filtration on probability space $(\Omega,\sigma)$, and assume the whole sigma algebra $\sigma$ is generated by all the $\mathcal{F}_n$. Does $\mathbb{E}(X|\mathcal{F}_n)$ converge pointwise to $X$? If not, can you give a simple example?
Is there any result on how well $E(X|\mathcal{G})$ approximates $X$, depending on how close $\mathcal{G}$ is to the whole sigma algebra $\sigma$?