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June 2006 A constructive view on ergodic theorems
Bas Spitters
J. Symbolic Logic 71(2): 611-623 (June 2006). DOI: 10.2178/jsl/1146620162

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.

Citation

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Bas Spitters. "A constructive view on ergodic theorems." J. Symbolic Logic 71 (2) 611 - 623, June 2006. https://doi.org/10.2178/jsl/1146620162

Information

Published: June 2006
First available in Project Euclid: 2 May 2006

zbMATH: 1106.03052
MathSciNet: MR2225897
Digital Object Identifier: 10.2178/jsl/1146620162

Rights: Copyright © 2006 Association for Symbolic Logic

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Vol.71 • No. 2 • June 2006
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