Journal of Symbolic Logic

A constructive view on ergodic theorems

Bas Spitters

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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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J. Symbolic Logic, Volume 71, Issue 2 (2006), 611-623.

First available in Project Euclid: 2 May 2006

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Spitters, Bas. A constructive view on ergodic theorems. J. Symbolic Logic 71 (2006), no. 2, 611--623. doi:10.2178/jsl/1146620162.

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