Define $M(Z_1, \dots, Z_n)$ to mean the closed subspace of $L^2$ consisting of all random variables in $L^2$ of the form $\phi(Z_1, \dots, Z_n)$ for some Borel function $\phi: R^n \rightarrow R$. Is it true for any $X, Y_1, Y_2, \dots , Y_n$ random variables that
$$E[X \ | \ \bar{\operatorname{sp}}\{Y_1, \dots, Y_n \} ] =E[X | M(Y_1, \dots , Y_n]$$
where $\bar{\operatorname{sp}}$ is the closed span of $Y_1, \dots, Y_n$ in $L^2$ ?
In general no, even when $n=1$. For example if $X=Y_1^2$, then $ E[X | M(Y_1, \dots , Y_n)]=X$ but in general $Y_1^2$ is not an element of the closure of the subspace generated by the constants and $Y_1$.