Hostname: page-component-cb9f654ff-lqqdg Total loading time: 0 Render date: 2025-08-22T16:03:35.672Z Has data issue: false hasContentIssue false
  • English
  • Français

The pointwise ergodic theorem in subsystems of second-order arithmetic

Published online by Cambridge University Press:  12 March 2014

Ksenija Simic
Ksenija Simic*
Affiliation:
Department of Mathematics, University of Arizona, 617 North Santa Rita, Tucson, AZ 85721, USA. E-mail: ksimic@math.arizona.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

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 ACA0; moreover, the pointwise ergodic theorem is equivalent to (ACA) over the base theory RCA0.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 1 , March 2007 , pp. 45 - 66
Copyright
Copyright © Association for Symbolic Logic 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Avigad, Jeremy and Reck, Erich, “Clarifying the nature of the infinite”: the development of metamathematics and proof theory, Technical Report CMU-PHIL-120, Carnegie Mellon University, 2001.Google Scholar
[2]Avigad, Jeremy and Simic, Ksenija, Fundamental notions of analysis in subsystems of second-order arithmetic, Annals of Pure and Applied Logic, vol. 139 (2006), pp. 138–184.CrossRefGoogle Scholar
[3]Billingsley, Patrick, Ergodic theory and information, Wiley, New York, 1965.Google Scholar
[4]Birkhoff, G. D., Proof of the ergodic theorem, Proceedings of the National Academy of Sciences, vol. 17 (1931), pp. 656–660.CrossRefGoogle ScholarPubMed
[5]Bishop, Errett, Foundations of constructive analysis, McGraw-Hill, New York, 1967.Google Scholar
[6]Brown, Douglas K., Giusto, Mariagnese, and Simpson, Stephen G., Vitali's theorem and WWKL, Archive for Mathematical Logic, vol. 41 (2002), no. 2, pp. 191–206.CrossRefGoogle Scholar
[7]Brown, Douglas K. and Simpson, Stephen G., Which set existence axioms are needed to prove the separable Hahn-Banach theorem?, Annals of Pure and Applied Logic, vol. 31 (1986), no. 2–3, pp. 123–144, special issue: second Southeast Asian logic conference (Bangkok, 1984).CrossRefGoogle Scholar
[8]Dunford, Nelson and Schwartz, Jacob T., Linear operators. Part I, Wiley Classics Library, John Wiley & Sons Inc., New York, 1988, Spectral operators, with the assistance of William G. Bade and Robert G. Bartle, reprint of the 1971 original, a Wiley-Interscience Publication.Google Scholar
[9]Frank, Matthew, Ergodic theory in the 1930s: a study in international mathematical activity, preprint.Google Scholar
[10]Garsia, A. M., A simple proof of Eberhard Hopf's maximal ergodic theorem, Journal of Applied Mathematics and Mechanics, vol. 14 (1965), pp. 381–382.Google Scholar
[11]Halmos, Paul R., Lectures on ergodic theory, Chelsea Publishing Co., New York, 1960.Google Scholar
[12]Humphreys, A. James and Simpson, Stephen G., Separable Banach space theory needs strong set existence axioms, Transactions of the American Mathematical Society, vol. 348 (1996), no. 10, pp. 4231–4255.CrossRefGoogle Scholar
[13]Kamae, Teturo, A simple proof of the ergodic theorem using nonstandard analysis, Israel Journal of Mathematics, vol. 42 (1982), no. 4, pp. 284–290.CrossRefGoogle Scholar
[14]Katznelson, Yitzhak and Weiss, Benjamin, A simple proof of some ergodic theorems, Israel Journal of Mathematics, vol. 42 (1982), no. 4, pp. 291–296.CrossRefGoogle Scholar
[15]Riesz, Frédéric, Sur la théorie ergodique, Commentarii Mathematici Helvetia, vol. 17 (1945), pp. 221–239.Google Scholar
[16]Shioji, Naoki and Tanaka, Kazuyuki, Fixed point theory in weak second-order arithmetic, Annals of Pure and Applied Logic, vol. 47 (1990), no. 2, pp. 167–188.CrossRefGoogle Scholar
[17]Simic, Ksenija, Aspects of ergodic theory in subsystems of second-order arithmetic, Ph.D. thesis, Carnegie Mellon University, 2004, draft available at http://math.arizona.edu/˜ksimic.Google Scholar
[18]Simic, Ksenija, Properties of Lp spaces in subsystems of second-order arithmetic, submitted, available at http://math.arizona.edu/˜ksimic.Google Scholar
[19]Simpson, Stephen G., Subsystems of second-order arithmetic, Springer, Berlin, 1998.Google Scholar
[20]Spitters, Bas, Constructive and intuitionistic integration theory and functional analysis, Ph.D. thesis, University of Nijmegen, 2002.Google Scholar
[21]Von Neumann, John, Proof of the quasi-ergodic hypothesis, Proceedings of the National Academy of Sciences, vol. 18 (1932), pp. 70–82.Google Scholar
[22]Walters, Peter, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York, 1982.CrossRefGoogle Scholar
[23]Yu, Xiaokang, Radon-Nikodým theorem is equivalent to arithmetical comprehension, Logic and computation (Pittsburgh, PA, 1987), Contemporary Mathematics, vol. 106, American Mathematical Society, Providence, RI, 1990, pp. 289–297.Google Scholar
[24]Yu, Xiaokang, Riesz representation theorem, Borel measures and subsystems of second-order arithmetic, Annals of Pure and Applied Logic, vol. 59 (1993), no. 1, pp. 65–78.CrossRefGoogle Scholar
[25]Yu, Xiaokang, Lebesgue convergence theorems and reverse mathematics, Mathematical Logic Quarterly, vol. 40 (1994), no. 1, pp. 1–13.CrossRefGoogle Scholar
[26]Yu, Xiaokang and Simpson, Stephen G., Measure theory and weak König's lemma, Archive for Mathematical Logic, vol. 30 (1990), no. 3, pp. 171–180.CrossRefGoogle Scholar