Intuitive explanation of the tower property of conditional expectation I understand how to define conditional expectation and how to prove that it exists.
Further, I think I understand what conditional expectation means intuitively. I can also prove the tower property, that is if $X$ and $Y$ are random variables (or $Y$ a $\sigma$-field) then we have that
$$\mathbb E[X] = \mathbb{E}[\mathbb E [X | Y]].$$
My question is: What is the intuitive meaning of this? It seems quite puzzling to me.
(I could find similar questions but not this one.)

Think of it as parallel to Bayes law on conditional probabilities.
the conditional expectations form a partition of the sample space of Y.
in discrete case bayes law says:
p(A)=p(A|B)p(B)+p(A|~B)p(B)
on the other hand:
p(A)=E(x)1_A
1_A being the indicator function of A.
