Suppose $\mathscr{C}_1\subset\mathscr{C}$ are two $\sigma$ fields, $X,Y$ are two random variables, and $E(XY|\mathscr{C})=E(X|\mathscr{C})\cdot E(Y|\mathscr{C})$, can we obtain that $E(XY|\mathscr{C}_1)=E(X|\mathscr{C}_1)\cdot E(Y|\mathscr{C}_1)$? And how to understand this intuitively?
This comes from a research problem. I am not sure about the answer. So I post it here for help.
How about if we add a condition on the random variable: suppose Y=D, where D is a binary random variable taking value of 0 and 1.