Nonexistence of measurable optimal selections

Abstract:

We give an example of a function $f$ on a separable metric space $X$ into a compact metric space $Y$ such that the graph of $f$ is a Borel subset of $X \times Y$, but $f$ is not Borel measurable. The example forms the basis for our construction of an upper semicontinuous, compact model of a one-day dynamic programming problem where the player has an optimal action at each state, but is unable to make a choice of such an action in a Borel measurable manner.