Abstract
The Banach–Zaretsky Theorem is a fundamental but often overlooked result that characterizes the functions that are absolutely continuous. This chapter presents basic results on differentiability, absolute continuity, and the Fundamental Theorem of Calculus with an emphasis on the role of the Banach–Zaretsky Theorem.
This work was partially supported by a grant from the Simons Foundation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Benedetto, J. J.: Real Variable and Integration. B. G. Teubner, Stuttgart (1976)
Benedetto, J. J., Czaja, W.: Integration and Modern Analysis. Birkhäuser, Boston (2009)
Bruckner, A. M., Bruckner, J. B., Thomson, B. S.: Real Analysis. Prentice Hall, Upper Saddle River, NJ (1997)
Cohn, D. L.: Measure theory. Birkhäuser, Boston (1980)
Folland, G. B.: Real Analysis, Second Edition. Wiley, New York (1999)
Heil, C.: Introduction to Real Analysis. Springer, Cham (2019)
Natanson, I. P.: Theory of Functions of a Real Variable. Translated from the Russian by Leo F. Boron with the collaboration of Edwin Hewitt. Frederick Ungar Publishing Co., New York (1955)
Rudin, W.: Real and Complex Analysis, Third Edition. McGraw-Hill, New York (1987).
Saks, S.: Theory of the Integral, Second revised edition. English translation by L. C. Young, Dover, New York (1964)
Serrin J., Varberg, D. E.: A general chain rule for derivatives and the change of variables formula for the Lebesgue integral. Amer. Math. Monthly 76, 514–520 (1969)
Stein, E. M., Shakarchi, R.: Real Analysis. Princeton University Press, Princeton, NJ (2005)
Varberg, D. E.: On absolutely continuous functions. Amer. Math. Monthly 72, 831–841 (1965)
Wheeden, R. L., Zygmund, A.: Measure and Integral. Marcel Dekker, New York-Basel (1977)
Acknowledgements
This chapter draws extensively from Chapters 5 and 6 of [6]. That material is used with permission of Springer. Many classic and recent volumes influenced the writing, the choice of topics, the proofs, and the selection of problems in [6]. We would like to explicitly acknowledge those tests that had the most profound influence. These include Benedetto and Czaja [2], Bruckner, Bruckner, and Thomson [3], Folland [5], Rudin [8], Stein and Shakarchi [11], and Wheeden and Zygmund [13]. I greatly appreciate all of these texts and encourage the reader to consult them. Many additional texts and papers are listed in the references to [6].
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Heil, C. (2021). Absolute Continuity and the Banach–Zaretsky Theorem. In: Hirn, M., Li, S., Okoudjou, K.A., Saliani, S., Yilmaz, Ö. (eds) Excursions in Harmonic Analysis, Volume 6. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-69637-5_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-69637-5_2
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-69636-8
Online ISBN: 978-3-030-69637-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)