= Dynamic programming operator
{title2=$\mathcal Lf(x)=\inf_u\{c(x,u)+\beta\mathbb E[f(X_1)\mid x,u]\}$}
This one-step operator combines immediate cost with optimized discounted continuation. It is monotone. For nonnegative costs, iterating it from zero produces increasing finite-horizon values bounded above by the infinite-horizon value. If a nonnegative fixed point has a minimizing measurable selector, repeated conditional expectation and dropping the nonnegative terminal value show that the corresponding stationary policy costs no more than that fixed point. Thus the infinite-horizon value, when the <Bellman equation> holds, is the least nonnegative fixed point.
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