Mixing processes in Riesz spaces and their ergodic properties

Abstract

The ergodic theorems of Hopf, Wiener and Birkhoff were extended to the context of Riesz spaces with a weak order unit and conditional expectation operator by Kuo, Labuschagne and Watson in [Ergodic Theory and the Strong Law of Large Numbers on Riesz Spaces. Journal of Mathematical Analysis and Applications,325, (2007), 422–437.]. However, the precise concept of what constitutes ergodicity in Riesz spaces was not considered. This omission will be filled in and some explanations of the choices made will be given. In addition, the interplay between mixing and ergodicity in the Riesz space setting is considered. In order for this to happen, the Koopman-von Neumann convergence condition on the Ces`aro mean must be be extended to the context of a Dedekind complete Riesz space with weak order unit. As a consequence, a characterisation of conditional weak mixing is given in the Riesz space setting. The results are applied to convergence in L1