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, > 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 ε, there is an ε-optimal strategy for the n-move game, measurable on the σ-field generated by the analytic sets.