$X,Y$ are random variables, and consider the following statements:
i) $X$ and $Y$ are independent, i.e. for $A\in \sigma(X)$ and $B\in\sigma(Y)$, $P(A\cap B) = P(A)P(B)$
ii) Mean Independence, $E(Y|X) = E(Y)$, and $E(X|Y) = E(X)$
iii) Uncorrelated, $E(XY)=E(X)E(Y)$
I know that i)$\implies$ii)$\implies$iii), and I know that the converse if false. However, in trying to get clear concepts, I hope to know what's missing in between. To be precise, my question is:
What's the condition $P(X,Y)$ and $Q(X,Y)$ such that
$X$ and $Y$ are independent $\iff X$ and $Y$ are independent in mean $\land P$
and
$X$ and $Y$ are independent in mean $\iff$ $X$ and $Y$ are uncorrelated $\land Q$
I don't know if this is exactly what you are looking for, but it may give you some insights:
Note that $E[g(Y)|X] = E[g(Y)]$ for any Borel measurable function $g$ implies independence of Y and X as $E[f(Y)h(X)] = E[f(Y)]E[h(X)]$ for any two Borel functions f and h.
On the other hand, conditional mean independence of Y and X implies that Y is uncorrelated with any Borel function of X: $E[Yg(X)] = E[Y]E[g(X)]$