Let $Y$ be a random variable and $\mathcal F_n$ be an increasing sequence of sigma algebras.
Do we have some limiting result for $E[Y \mid \mathcal F_n]$?
For example, if $$\mathcal F_\infty = \limsup_{n \to \infty} \mathcal F_n = \cap_{n=1}^\infty \cup_{m\ge n} \mathcal F_m$$
is a sigma algebra, can we say that $$\lim_{n \to \infty} E[Y \mid \mathcal F_n] = E[Y \mid \mathcal F_\infty]$$?