Let $X,Y$ be bounded random variables on $(\Omega, \mathcal{F}, \mathbb{P})$. I want to show that each implication implies the next, but the converses are all false:
(i) $X, Y$ are independent
(ii) $\mathbb{E}[X|Y] = \mathbb{E}[X]$ a.s.
(iii) $\mathbb{E}[XY] = \mathbb{E}[X] \mathbb{E}[Y]$
I find it confusing that we have the statement (ii) in the middle, and that's what is causing me trouble to figure out the proofs and counterexamples.