Let $X: \Omega \rightarrow [0,1]^K$ be a random vector. Further, we assume that there exist a random vector $Z$ such that $\mathbb{E}[Z|X = x] = 0$ for all $x$ in the support of $X$.
Given this assumption is it also true that we have $\mathbb{E}[Z_k|X_k = x_k] = 0$ for all $k$? Obviously, we have that $\mathbb{E}[Z_k| X = x] = 0$ for all $x$ in the support of $X$, but I don't know how to prove (or disprove) the claim. So far I tried using law of total expectation, and various other approaches, but I could not reach a conclusive solution (without assuming more than the given condition).
Any help is highly appreciated.
For $X,Y,Z$ random vectors, if $E[Z\mid X,Y]=0$ we have: \begin{gather*} E[Z\mid X]=\int E[Z\mid X,Y=y]dP_{Y/X}=\int 0dP_{Y/X}=0 \end{gather*}