The Optimal Reward Operator in Dynamic Programming

Consider a dynamic programming problem with analytic state space S, analytic constraint set A, and semi-analytic reward function r (x, P, y) for (x, P) A and y S: namely, \r > a\ is an analytic set for all a. Let Tf be the optimal reward in one move, with the modified reward function r (x, P, y) + f (y). The optimal reward in n moves is shown to be Tⁿ0, a semi-analytic function on S. It is also shown that for any n and positive, there is an -optimal strategy for the n-move game, measurable on the -field generated by the analytic sets.