, Volume 6, Issue 5, pp 965-985
Date: 02 Aug 2011

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

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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 L p (ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L p (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 L p (ℰ(E,H) the complete random normed module generated by L p (ℰ, H).