Let $X \in L^1$ and $Y \in L^\infty$ be two random variables (or, more generally, $X \in L^p, \, Y \in L^q$ for $p, \, q$ conjugated). Is it true that
$$ \mathbb{E} \left[ X Y \right] = \sup_{ \hat{Y} \sim Y} \mathbb{E} \left[ X \hat{Y} \right]$$
where the supremum is taken over all $\hat{Y}$ that have the same distribution as $Y$?
Clearly, the right-hand side is well-defined, but could a-priori be $\infty$. Also, the right-hand side is clearly larger than the left.
How could one prove the other direction $\geq$?
False. Let $Y$ have standard normal distribution and $X=-Y$. Then $EXY=-1$ but $\hat {Y} =X$ has the same distribution as $Y$ so RHS $\geq EX^{2}=1$. This answers the second part. For the first part you need a bounded random variable, so replace normal distribution by any symmetric bounded distribution.