Global Upper Expectations for Discrete-Time Stochastic Processes: In Practice, They Are All The Same!

We consider three different types of global uncertainty models for discrete-time stochastic processes: measure-theoretic upper expectations, game-theoretic upper expectations and axiomatic upper expectations. The last two are known to be identical. We show that they coincide with measure-theoretic upper expectations on two distinct domains: monotone pointwise limits of finitary gambles, and bounded below Borel-measurable variables. We argue that these domains cover most practical inferences, and that therefore, in practice, it does not matter which model is used.