When will $E(X\mid Y)=EX$ be true? A maybe easier question:
Can you find an example that satisfies

$X$ and $Y$ are not independent, but $E(X\mid Y) = EX$.


My actual question:
In what cases (other than independence between $X$ and $Y$), will $E(X\mid Y)=EX$ be true? 
The weaker the case, the better.
Thanks!
 A: Suppose you want the expected value of $X$ to be some particular number $\mu$.  For every value that the random variable $Y$ could take, pick some probability distribution for which the expected value exists.  If $W$ is a random variable with that distribution, then let $W-\mathbb EW + \mu$ be the value of $X$ when $Y$ assumes the particular value concerned.  Then $\mathbb E(X\mid Y) =\mu$ with probability $1$, and $\mathbb E(X)=\mu$.  But $X$ and $Y$ are far from independent if you picked a different distribution for each value that $Y$ could take.
Concrete example: Say you want $\mathbb EX$ to be $6$.  Say $\displaystyle Y=\begin{cases} 0 & \text{with probability }1/2, \\ 1 & \text{with probability }1/2. \end{cases}$  If $Y=0$, let $X$ be $6$ plus a standard normal random variable.  If $Y = 1$, let $X$ be $6\pm1$, each with probability $1/2$.  Then $\mathbb E(X\mid Y)=6$ with probability $1$ and $\mathbb E(X)=6$, but $X$ and $Y$ are far from being independent.
A: In general, the conditional expectation is a function of the conditioning variable $E(X\mid Y) =g(Y)$  When this function is a constant, then (iff) $E(X\mid Y) =E(X)$. You can't say much more, in general.
Here I give an example.
