Conditional Essential Suprema with Applications

The conditional supremum of a random variable X on a probability space given a sub--algebra is defined and proved to exist as an application of the Radon–Nikodym theorem in L ^\infty. 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 ^\infty 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.