Is it possible to remove a variable from a conditional expectation expression?

Question

Is it possible to remove F=f(x,y) from the following expression:

E(E(Z|F,X,Y) | F,X)

i.e. it becomes,

E(E(Z|X,Y) | F,X)

which then becomes,

E(Z|X,Y)

??

Answer 1

Actually, the first step (where you removed the F) is correct. It's the second step that's problematic.

Notice that,

$$\sigma(F) \subseteq \sigma(X,Y)$$

So, $\sigma(F,X,Y) = \sigma(X,Y)$. That's why,

$$E(Z|F,X,Y) = E(Z|X,Y).$$

However, if there exist $x$ such that $f(x,\cdot)$ is not a one-to-one function, then $\sigma(X,F) \subsetneq \sigma(X,Y)$. If these sigma algebras differ by some set of non-trivial probability, then

$$E(E(Z|X,Y)|X,F) = E(Z|X,F) \neq E(Z|X,Y).$$