Convergence of iterated conditional expectations

Question

Notation: We write $\mathbb E_{\mathcal F} X$ for the conditional expectation $\mathbb E[X|\mathcal F]$ of a random variable $X$ with respect to a $\sigma$-algebra $\mathcal F$.

Let $X$ be an integrable random variable with associated $\sigma$-algebra $\mathcal X$, and let $\mathcal G, \mathcal H$ be sub $\sigma$-algebras of $\mathcal X$. It is known that the seemingly intuitive identity

$$\mathbb E_{\mathcal G} \mathbb E_{\mathcal H} X = \mathbb E_{\mathcal H} \mathbb E_{\mathcal G} X = \mathbb E_{\mathcal G \cap \mathcal H} X$$

does not hold in general, see for example here for an elementary construction where all three of the values above are different.

However, I believe the following holds.

Question: Is it true that the sequence

$$\mathbb E_{\mathcal H} X, \mathbb E_{\mathcal G} \mathbb E_{\mathcal H} X, \mathbb E_{\mathcal H} \mathbb E_{\mathcal G} \mathbb E_{\mathcal H} X, \mathbb E_{\mathcal G} \mathbb E_{\mathcal H} \mathbb E_{\mathcal G} \mathbb E_{\mathcal H} X, \dots$$

converges almost surely to $ \mathbb E_{\mathcal G \cap \mathcal H} X$?

Comments: The desired result seems to be true if $X$ is a random variable taking finitely many values. Indeed, viewing the conditional expectation operator as a projection, the convergence is guaranteed by the following algorithm, popularised by von Neumann.

Answer 1

Here is a partial answer: It is at least true under the slightly stronger integrability assumption that $X \in L^p$ for some $p > 1$ (I don't know about the case $X \in L^1$, though).

First, as mentioned in the comments, strong convergence on $L^2$ (on hence on $L^1$ by density) follows from a Theorem a von Neumann for alternating orthogonal projections on Hilbert spaces.

The almost everywhere convergence for $X \in L^p$ with $p > 1$ follows from the following result of Stein (see Corollary 1 on page 1895 in this classical article):

Theorem. Let $(\Omega,\mu)$ be a measure space and let $P$ be a self-adjoint and positive semi-definite bounded linear operator on $L^2(\Omega,\mu)$. Assume that $P$ extrapolates to a linear operator of norm $\le 1$ on $L^1$. Then $P^nf$ converges almost everywhere as $n \to \infty$ if $f \in L^p(\Omega,\mu)$ for some $p \in (1,\infty)$.

Now, to obtain the claim for $X \in L^p$ with $p > 1$, apply the theorem to each of the two operators $P_1 := \mathbb{E}_{\mathcal{H}} \mathbb{E}_{\mathcal{G}} \mathbb{E}_{\mathcal{H}}$ and $P_2 := \mathbb{E}_{\mathcal{G}} \mathbb{E}_{\mathcal{H}} \mathbb{E}_{\mathcal{G}}$ and use von Neumann's theorem to see that the almost everywhere limits of $P_1^n X$ and $P_2^n \mathbb{E}_{\mathcal{H}} X$ coincide.