Let $\mathcal{A}_n$ be a sequence of finite sigma-algebras, let $\mathcal{B}_{q,p}= \sigma(\mathcal{A}_n, q \geq n \geq p )$. Moreover, we suppose $\mathcal{A}_k \subset \mathcal{B}_{\infty,p}$ for any $k <p$.

Let $X$ be a random variable that is $\mathcal{B}_{\infty,p}$ measurable for any integer $p$.

Let $u_{q,p}= E[ |E[X|\mathcal{B}_{q,p}]-E[X|\mathcal{A}_q] |^2]$.

We suppose for any $p \geq 1$, $u_{q+1,p}=u_{q,p-1}$ (this is a weaker equivalent of the "filtration" condition in the martingale convergence theorem) , and that $u_{n+1,0}-u_{n,0} \rightarrow_{n \rightarrow + \infty} 0$.

Is it true that almost surely, $E[X|\mathcal{A}_n] \rightarrow_{n \rightarrow + \infty} X$.

If not, can you give a counterexample?