# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] An introduction to ergodic theory 79 (1982) [2] Logic and computation (Pittsburgh, PA, 1987) 106 pp 289–297– (1990) [3] Linear operators. Part I (1988) [4] Annals of Pure and Applied Logic 31 pp 123–144– (1986) [5] Archive for Mathematical Logic 41 pp 191–206– (2002) [6] Foundations of constructive analysis (1967) · Zbl 0183.01503 [7] Proceedings of the National Academy of Sciences 17 pp 656–660– (1931) [8] Ergodic theory and information (1965) [9] Annals of Pure and Applied Logic 139 pp 138–184– (2006) [10] ”Clarifying the nature of the infinite”: the development of metamathematics and proof theory (2001) [11] Subsystems of second-order arithmetic (1998) [12] Annals of Pure and Applied Logic 47 pp 167–188– (1990) [13] Commentarii Mathematici Helvetia 17 pp 221–239– (1945) [14] Israel Journal of Mathematics 42 pp 291–296– (1982) [15] Israel Journal of Mathematics 42 pp 284–290– (1982) [16] Transactions of the American Mathematical Society 348 pp 4231–4255– (1996) [17] Lectures on ergodic theory (1960) [18] Journal of Applied Mathematics and Mechanics 14 pp 381–382– (1965) [19] Archive for Mathematical Logic 30 pp 171–180– (1990) [20] Mathematical Logic Quarterly 40 pp 1–13– (1994) [21] Annals of Pure and Applied Logic 59 pp 65–78– (1993) [22] Proceedings of the National Academy of Sciences 18 pp 70–82– (1932)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.