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

$E[A|Y]=A$ if $A$ is $\sigma(Y)$-measurable.
Just choose $A=E[X|Z]$. Remember conditional expectation is a random variable.
