How to geometrically interpret conditional expectation property tower rule?? I have a question on how to interpret conditional expectation its properties geometrically.  There are two properties of conditional expectation in particular that I’m trying to interpret:
Given a probability space $(\Omega, \mathcal{G}, \mathbb{P})$ 


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*If $\mathcal{H} \subset \mathcal{G}$ then $\mathbb{E}[\mathbb{E}[X|\mathcal{H}]  | \mathcal{G}] = \mathbb{E}[ X|\mathcal{H}]$.  (tower rule)

*If $ X \mathrel{\unicode{x2AEB}} G$ then $\mathbb{E}[X|\mathcal{G}] = \mathbb{E}[ X]$.
I interpret the tower rule geometrically using linear algebra as follows: 


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*$H \subset G$ is interpreted as $\mathcal{H}$ is a subspace contained in the subspace $\mathcal{G}$.

*$\mathbb{E}[X|\mathcal{H}]$ is the projection of $X$ onto $\mathcal{H}$.

*So, $\mathbb{E}[\mathbb{E}[X|\mathcal{H}]  | \mathcal{G}] = \mathbb{E}[ X|\mathcal{H}]$ can be interpreted as the projection of X onto $\mathcal{H}$, projected onto $\mathcal{G}$, is the same as the projection of X onto $\mathcal{H}$.
This makes sense.  However, the reason I don't like this interpretation is that it is also true if I swap $\mathcal{H}$ and $\mathcal{G}$.  So, this doesn't use the property that $\mathcal{H} \subset \mathcal{G}$.  Can someone help me come up with a better interpretation of this?
related: Question about conditional expectation as projection
 A: Let's start with a little Linear Algebra warm-up.
Let $V$ be a vector space (which might be infinite dimensional, in that case we will require it to be a Hilbert space), and let $W \subset U$ be two (closed) sub-spaces of $V$.
Denote by $P_W$ and $P_U$ the orthogonal projections on $W$ and $U$, respectively.
It is straightforward to check
$$P_W\cdot P_U = P_U\cdot P_W = P_W.$$
That is, the operators commute and their product is precisely $P_W$.
Back to the probabilistic setting. We consider the Hilbert space $L^2(\mathcal{G})$ of square integrable random variables measureable with respect to $\mathcal{G}$. Now suppose that there exists $\mathcal{H}_1 \subset \mathcal{H}_2 \subset \mathcal{G}$, and consider the closed subspaces $L^2(\mathcal{H}_1)$, $L^2(\mathcal{H}_2)$ of square integrable random variables with respect to $\mathcal{H}_1$ and $\mathcal{H}_2$.
It is clear that $L^2(\mathcal{H}_1) \subset L^2(\mathcal{H}_2)$, and by the previous comment 
$$P_{L^2(\mathcal{H}_1)}\cdot P_{L^2(\mathcal{H}_2)} = P_{L^2(\mathcal{H}_2)}\cdot P_{L^2(\mathcal{H}_1)}= P_{L^2(\mathcal{H}_1)}.$$
Now, we can interpert $P_{L^2(\mathcal{H}_1)}(X)$ as $\mathbb{E}\left[X|\mathcal{H}_1\right]$ and $P_{L^2(\mathcal{H}_2)}(X)$ as $\mathbb{E}\left[X|\mathcal{H}_2\right]$ to get the desired result.
