revealed preference given an uncertain environment? 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 $X$.
Revealed preference theory is essentially about the observable empirical content of theories. Now if we look at the classical model of utility maximization, if there is a fixed function $f:X\to O$ and a utility function $u:O\to\mathbb{R}$, utility maximization amounts to maximizing with respect to the utility function $v:X\to\mathbb{R}$ given by $v(x)=u\big(f(x)\big)$ for all $x\in X$. There is nothing new here.

If $f$ can change or is uncertain, you are looking at a form of changing preferences or uncertain preferences. Since revealed preference theory is about the observable empirical content of theories, you need some theory about changing preferences or uncertain preferences. The task is then to identify the empirical content of the theory. For this to be an interesting exercise, there needs to be some relation between different decision problems. One example is the work of Gul and Pesendorfer on the empirical implications of the Strotz multiple self model of changing preferences with consistent planning in


  Gul, Faruk, and Wolfgang Pesendorfer. "The revealed preference theory of changing tastes." The Review of Economic Studies 72.2 (2005): 429-448.