A Constructive View on Ergodic Theorems

Journal of Symbolic Logic 71 (2):611 - 623 (2006)
Abstract
Let T be a positive L₁-L∞ contraction. We prove that the following statements are equivalent in constructive mathematics. (1) The projection in L₂ on the space of invariant functions exists: (2) The sequence (Tⁿ)n∈N Cesáro-converges in the L₂ norm: (3) The sequence (Tⁿ)n∈N Cesáro-converges almost everywhere. Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem. As a corollary we obtain a constructive ergodic theorem for ergodic measure-preserving transformations. This answers a question posed by Bishop
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