Let $(A,\mathcal{A},P)$ be a probability space, $f$ real, bounded, nonnegative, measurable function and $X,Y$ real random variables so that $E(Y)=0$.
Then $E(f(X)Y)=0$.
My arguments
$$0=-C\int YdP\leq\int -f(X)YdP\leq\int f(X)YdP\leq C\int YdP=0,$$ for some constant $C>0$.
*I just realize that the second inequlity does not need to hold.
Can you give me feedbacks about it?
Thank you in advance!