Independence between conditional expectations Suppose $(\Omega, F, P)$ is a sample space, $X$ and $Y$ random variables, and $N$ and $M$ sub sigma algebras of $F$. 


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*I know that $E(X\mid N)$ and $E(X\mid\{\emptyset, \Omega\})$ are independent. Generally, if $N$ and $M$ are independent, will $E(X\mid N)$ and $E(X\mid M)$ be independent? What are some conditions on $N$, $M$ and $X$ that can make $E(X\mid N)$ and $E(X\mid M)$  independent?

*If $X$ and $Y$ are independent, are $E(X\mid N)$ and $E(Y\mid N)$ independent? What are some conditions on $N$, $X$ and $Y$ that can make $E(X\mid N)$ and $E(Y\mid N)$  independent?
Thanks.

To answer your second question: Let $X, Y$ be independent random variables that assume the values $0$ and $1$ each with probability $\frac{1}{2}$. Setting $N = X + Y$ you get
$$E[X \mid X + Y] = E[Y \mid X + Y] = \frac{X + Y}{2}.$$
