Show that F(X) and Y-E[Y|X] uncorrelated

Question

X, Y are random variables on probability space , f is a measurable function, Show that F(X) and Y-E[Y|X] uncorrelated.

I have tried the followings :

Cov(f(X),Y-E[Y|X]) = E[f(X)(Y-E[Y|X])] -E[f(X)]E[Y-E[Y|X]]

= E[f(X)Y]-E[f(X)E[Y|X]]-E[f(X)].E[Y]+E[f(X)].E[E[Y|X]]

, due to E[Y]=E[E[Y|X]], then the above reduce to

= E[f(X)Y]-E[f(X)E[Y|X]]

at this step I stuck, so I really appreciate if anyone can give the trick to complete this problem.

Answer 1

Hint: Push $f(X)$ inside the conditional expectation as it is $\sigma(X)$-measurable and re-use the law of iterated expectations.

Answer 2

$$ E[f(X)Y] = E[E[f(X)Y|X] = E[ f(X) \cdot E[Y|X]]$$