This thesis consists of three self-contained essays and a concluding chapter. The summary of the main results of the essays is the following: First, we prove a Donsker result for the G-Brownian motion with finite state-space.

In the second essay, we give an elementary and more intuitive introduction to nonstandard measure theory and we also provide an alternative construction of the renowned Loeb measure. Following, we develop the basic theory for the hyperfinite G-expectation.