The Kac formula and Poincar\'{e} recurrence theorem in Riesz spaces

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 Poincar\'{e} 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 Ces\`aro mean associated with the iterates of the process. Applied to processes in $L^1(\Omega,{\mathcal A},\mu)$, where $\mu$ is a probability measure, new conditional versions of the above theorems are obtained.