Consider two random variables $X$ and $Y$. Let $Z_1,Z_2$ be two random variables, measurable with respect to the $\sigma$-field generated by $X,Y$ such that $$\mathbb E (X\mid Z_1)=E (X\mid Z_2)$$ $$\mathbb E (Y\mid Z_1)=E (Y\mid Z_2)$$
What can I say about $Z_1$ and $Z_2$? Do they generate the same $\sigma$-algebra? Does there exist a one-to-one function $f$ such that $Z_1=f(Z_2)$? Thanks!