The Kac formula and Poincaré recurrence theorem in Riesz spaces

Abstract

Riesz space (non-pointwise) generalizations for iterative processes are given for the concepts of recurrence, first recurrence and conditional ergodicity. Riesz space conditional versions of the Poincaré Recurrence Theorem and the Kac formula are developed. Under mild assumptions, it is shown that every conditional expectation preserving process is conditionally ergodic with respect to the conditional expectation generated by the Cesàro mean associated with the iterates of the process. Applied to processes in L 1 ( Ω , A , μ ) L^1(\Omega ,{\mathcal A},\mu ) , where μ \mu is a probability measure, new conditional versions of the above theorems are obtained.