conditional expectation and independence. Let $X_1, X_2,Y_1, Y_2$ be random variables and $F$ be subsigma field. 
Suppose that $X_i$ is measurable on $F$, $Y_1$ and $Y_2$ are independent given $F$. Then is this ture? $E(f(X_1,Y_1)g(X_2,Y_2)|F)=E(f(X_1,Y_1)|F)E(g(X_2,Y_2)|F)$
This is false even in the simplest set up when $\mathcal{F}$ is the entire trivial $\sigma$-algebra of the probability space. In this case, conditional expectation is just normal expectation. Let $Y_1$ and $Y_2$ be two independent coin flip ($\pm 1$ with probability $1/2$) and set $X_1 = Y_2$, $X_2 = Y_1$, $f(x,y) = g(x,y) = x+y$. Then LHS $= 2$ but RHS $= 0$.
This is false even in the simplest set up when $\mathcal{F}$ is the entire trivial $\sigma$-algebra of the probability space. In this case, conditional expectation is just normal expectation. Let $Y_1$ and $Y_2$ be two independent coin flip ($\pm 1$ with probability $1/2$) and set $X_1 = Y_2$, $X_2 = Y_1$, $f(x,y) = g(x,y) = x+y$. Then LHS $= 2$ but RHS $= 0$.