Suppose we have a natural filtration $\{\mathcal{F}_n\}$ of a stochastic process $\{X_n\}$ and an event $A$ in a supporting probability space $(\Omega,\mathcal{F},\mathbb{P})$. Often certain distributional properties of the process are stated in the following form: $\mathbb{P}(A|\mathcal{F}_n)=f(X_1,\ldots,X_n)$ on some event $E$, which is measurable with respect to $\mathcal{F}_n$.
My question is: what is the intuition behind such type of proposition, in particular behind evaluating the conditional probability on an event. We know that for example this has a very specific meaning when the filtration is generated by a partition and the event is in one of the partitioning sets. In this case the evaluation turns the conditional probability into an elementary conditioning on the part of the partition containing the event $E$. But what about the more general case, when the sigma-algebra is not generated by a partition? Is there still a heuristic way to understand the evaluation of the conditional probability on the event $E$?