I'm learning about mean independence and was wondering if the following is true:
Given R.V. X and Y, if E[X|Y] = 0, then since E[x] = E[E[X|Y]] = E[0] = 0. In other words, given two random variables such that the expectation of the first conditioned on the second is zero, then the first variable is mean independent on the second.
Yes, this is true, provided $E[X]$ exists. But it could be that $E[X]$ does not exist. For example, suppose $Y$ has uniform distribution on $(0,1]$, and given $Y=y$, $X$ takes values $1/y$ with probability $1/2$ and $-1/y$ with probability $1/2$.