We provide extension procedures for nonlinear expectations to the space of all bounded measurable functions. We first discuss a maximal extension for convex expectations which have a representation in terms of finitely additive measures. One of the main results of this article is an extension procedure for convex expectations which are continuous from above and therefore admit a representation in terms of countably additive measures. This can be seen as a nonlinear version of the Daniell–Stone theorem. From this, we deduce a robust Kolmogorov extension theorem which is then used to extend nonlinear kernels to an infinite-dimensional path space. We then apply this theorem to construct nonlinear Markov processes with a given family of nonlinear transition kernels.
Robert Denk. Michael Kupper. Max Nendel. "Kolmogorov-type and general extension results for nonlinear expectations." Banach J. Math. Anal. 12 (3) 515 - 540, July 2018. https://doi.org/10.1215/17358787-2017-0024