Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X$ satisfying $\mathbf{E}[|X|]<\infty$.

Define the *iterated expectations* of *X* as follows: $X_0 = X$, and, inductively, $X_k = \mathbf{E}[X_{k-1} \, | \, \mathcal{G}_k]$, where $\mathcal{G}_k \subseteq \mathcal{F}$ is some $\sigma$-algebra. Assume that all the $\mathcal{G}_k$ come from some finite set of $\sigma$-algebras.

In words, we are repeatedly taking conditional expectations with respect to various information.

**Can we conclude that the sequence $(X_k)_k$ converges $\mathbf{P}$-almost surely?**

In my comment on the question I include some of what I know about this, but I suspect that this is easy if one looks at it right, or covered in a standard reference.

finiteprobability spaces, irrespective of nestedness, I'd be surprised if it became a problem only once we go to general probability spaces. $\endgroup$1more comment