Conditional essential suprema with applications

The conditional supremum of a random variable X on a probability space given a sub-sigma-algebra is defined and proved to exist as an application of the Radon-Nikodym theorem in L-infinity. After developing some of its properties we use it to prove a new ergodic theorem showing that a time maximum is a space maximum. The concept of a maxingale is introduced and used to develop the new theory of optimal stopping in L-infinity and the concept of an absolutely optimal stopping time. Finally, the conditional max is used to reformulate the optimal control of the worst-case value function.