  Kolmogorovs axiomatic framework is the best-known approach to describing
probabilities and, due to its use of the Lebesgue integral, leads to remarkably
strong continuity properties. However, it relies on the specification of a
probability measure on all measurable events. The game-theoretic framework
proposed by Shafer and Vovk does without this restriction. They define global
upper expectation operators using local betting options. We study the
continuity properties of these more general operators. We prove that they are
continuous with respect to upward convergence and show that this is not the
case for downward convergence. We also prove a version of Fatous Lemma in this
more general context. Finally, we prove their continuity with respect to
point-wise limits of two-sided cuts.
