Subjective expected utility representations for Savage preferences on topological spaces

In many decisions under uncertainty, there are technological constraints on both the acts an agent can perform and the events she can observe. To model this, we assume that the set S of possible states of the world and the set X of possible outcomes each have a topological structure. The only feasible acts are continuous functions from S to X, and the only observable events are regular open subsets of S. In this environment, we axiomatically characterize a Subjective Expected Utility (SEU) representation of preferences over acts, involving a continuous utility function on X (unique up to positive affine transformations), and a unique probability measure on a Boolean algebra B of regular open subsets of S. With additional topological hypotheses, we obtain a unique Borel probability measure on S, along with an auxiliary apparatus called a liminal structure, which describes the agent’s informational constraints. We also obtain SEU representations involving subjective state spaces, such as the Stone-Čech compactification of S and the Stone space of B.