Law of Iterated Expectations applied thrice

Question

If $U(Z(Y,X),Y(X),X)$, then

$$E(U)= E(E(E(E(U\mid Z)\mid Y)\mid X))$$

Will this hold?

I think so it should hold because if we assume $Z$ is constant, then were assuming $X$ and $Y$ are constant and similarly for $Y$ and $X$.

Answer 1

Yes, but they should be iterated expectations, so the inner one should be conditional on $X,Y,Z$, the middle one on $X,Y$, the outer one on $X$ only (I think this is what you mean by $Y\left(X\right)$, etc.). Each time you are providing additional information on top of what was already known. Then, all of the usual properties of expectation carry over to conditional expectations.