Von Neumann’s mean ergodic theorem on complete random inner product modules

Abstract

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on Lℱp(ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, Lp(ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and Lℱp(ℰ(E,H) the complete random normed module generated by Lp(ℰ, H).