The pointwise ergodic theorem in subsystems of second-order arithmetic.

*(English)*Zbl 1116.03056Summary: The pointwise ergodic theorem is nonconstructive. In this paper, we examine origins of this non-constructivity, and determine the logical strength of the theorem and of the auxiliary statements used to prove it. We discuss properties of integrable functions and of measure preserving transformations and give three proofs of the theorem, though mostly focusing on the one derived from the mean ergodic theorem. All the proofs can be carried out in ACA\(_0\); moreover, the pointwise ergodic theorem is equivalent to ACA over the base theory RCA\(_0\).

##### MSC:

03F35 | Second- and higher-order arithmetic and fragments |

03B30 | Foundations of classical theories (including reverse mathematics) |

28D05 | Measure-preserving transformations |

##### Keywords:

pointwise ergodic theorem; logical strength; integrable functions; measure preserving transformations
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