Suppose the random variables X and Y follow an unknown joint distribution, and we do not know the independence of X and Y.
If I want to prove, let's assume $\mathbb{E}[XY]$ < $\mathbb{E}[XY^2]$, is it equivalent to prove by taking the conditional expectation that: for $\forall x \in X(w): \mathbb{E}[XY | X=x] < \mathbb{E}[XY^2|X=x]$?
If the latter one can prove the former one, is the latter one necessary?