Let $X,Y$ random variables with $Y$ real valued. I was wondering if any inequality of the type: \begin{equation} \mathbb{E}[f(X,Y)] \leq g \left[\sup_{\displaystyle s \in \mathbb{R}} \mathbb{E}[f(X,s)] \right] \end{equation}
exists, where $g$ is some function. It seems very wrong at first sight, but I was wondering if you knew something of this type.