Is the relation E[E[X|Z]|Y ] = E[X|Z] true or False? Given, we do not know if X,Y,Z are independent or not. Will the expression E[E[X|Z]|Y] = E[X|Z] hold true?
What I have tried:
Now, E[X|Z] is a random variable so let E[X|Z] be A
So, problem boils down to E[A|Y] which can only be equal to E[A] if A and Y are independent.
I would like to know if my thought process is accurate or not

If A and Y are independent, then E[E[X|Z]|Y] = E[X], as

*

*E[E[X|Z]|Y] = E[E[X|Z]] (property of conditional expectation: conditioning on independent factor), and

*E[E[X|Z]] = E[X] (law of iterated expectations).

If Z is dependent on Y (e.g., Z = 2*Y), I saw in Martingales introduction, they swap the two factors and further simplify: E[E[X|Z]|Y]=E[E[X|Y]|Z]=E[X|Z]. However, in that example where this was used, Y and Z are sigma-algebras, so I don't know whether this can be done with general dependent random variables. Otherwise, I am not sure, whether you can actually simplify it.
