  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. In this short paper
we fill in this omission and give some explanations of the choices made. In
addition, we consider the interplay between mixing and ergodicity in the Riesz
space setting.

Dynamical Systems;Functional Analysis;Probability