I have a random variable $X$ and its conditional expectation $E(X | G) = Y$.
In the context of some homework, I have to reach the conclusion that $P(X = 0 \text{ and } Y \neq 0) = 0$ But isn't this always the case? I think we have $P(X = 0 \text{ and } Y \neq 0) = P(Y \neq 0 | X=0) P(X=0)$ and $P(Y \neq 0 | X=0) = 0$ since when $X = 0$, its conditional expectation is $0$ too
Am I missing something here? thanks!