Nonexistence of measurable optimal selections

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 × 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.