In the recent surge of papers on ergodic theory within Riesz spaces, this
article contributes by introducing enhanced characterizations of ergodicity.
Our work extends and strengthens prior results from both the authors and
Homann, Kuo, and Watson. Specifically, we show that in a conditional
expectation preserving system (E,T,S,e), S can be extended to the natural
domain of T and operates as an isometry on L^{p}(T) spaces.Comment: 14 page