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The pointwise ergodic theorem in subsystems of second-order arithmetic. (English) Zbl 1116.03056
Summary: 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
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