classical revealed preference is about a situation where we assume an agent has a preference over some set of possible choices $X$. We then construct a revealed preference relation on $X$ from choices made given choice sets $C\subseteq X$.
But is there a theory of revealed preference for the case where the agent doesn't inherently care about his choice $x\in X$, but cares about outcome $o\in O$, where there is a function $f:X\to O$? This function $f$ may possibly be uncertain, or may vary over time?
EDIT: I am asking for a literature on the question of how we infer $u$ in this context, not on how do we infer the induced preference on $X$.