Is there any relationship between conditional expectations and the expectation of the product of random variables?

Question

My question is the following:

Given two generic R.V. $X$ and $Y$, is there any closed form formula for the relationship between the conditional expectation $E(X|Y)$ and the expectation of the product of the two R.V. $E(XY)$?

In particular, if $E(X|Y)=0$ could I conclude that $E(XY)=0$, as well? If not, what are the additional assumptions needed in order to say that if $E(X|Y)=0$ than $E(XY)=0$? Thank you.

Answer 1

$E(XY)=E(E(XY|Y)=E(YE(X|Y))=0$ provided $XY$ has finite expectation.

Answer 2

the law of total expectation

https://en.wikipedia.org/wiki/Law_of_total_expectation

$E(E(X|Y))=E(X)$

so if $E(X|Y)=0$

$E(XY)=E(E(XY|Y))=E(YE(X|Y))=E(Y\times 0)=0$

conditions: $E(|X|)< \infty$ $E(|XY|)<\infty$