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Notes on Real Analysis and Measure Theory

Fine Properties of Real Sets and Functions

  • Book
  • © 2022

Overview

Authors:
  1. Alexander Kharazishvili

    1. Andrea Razmadze Mathematical Institute, Tbilisi State University, Tbilisi, Georgia

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  • This monograph gives an up-to-date account of the fine properties of real sets and functions
  • Stresses the relationship between real analysis and descriptive set theory
  • Gives the reader a self-contained introduction to the unifying notion of non-measurability

Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xi
    Download chapter PDF

  2. Real-Valued Semicontinuous Functions

    • Alexander Kharazishvili
    Pages 1-19

  3. The Oscillations of Real-Valued Functions

    • Alexander Kharazishvili
    Pages 21-35

  4. Monotone and Continuous Restrictions of Real-Valued Functions

    • Alexander Kharazishvili
    Pages 37-49

  5. Bijective Continuous Images of Absolute Null Sets

    • Alexander Kharazishvili
    Pages 51-64

  6. Projective Absolutely Nonmeasurable Functions

    • Alexander Kharazishvili
    Pages 65-75

  7. Borel Isomorphisms of Analytic Sets

    • Alexander Kharazishvili
    Pages 77-92

  8. Iterated Integrals of Real-Valued Functions of Two Real Variables

    • Alexander Kharazishvili
    Pages 93-103

  9. The Steinhaus Property, Ergodicity, and Density Points

    • Alexander Kharazishvili
    Pages 105-121

  10. Measurability Properties of H-Selectors and Partial H-Selectors

    • Alexander Kharazishvili
    Pages 123-141

  11. A Decomposition of an Uncountable Solvable Group into Three Negligible Sets

    • Alexander Kharazishvili
    Pages 143-157

  12. Negligible Sets Versus Absolutely Nonmeasurable Sets

    • Alexander Kharazishvili
    Pages 159-168

  13. Measurability Properties of Mazurkiewicz Sets

    • Alexander Kharazishvili
    Pages 169-182

  14. Extensions of Invariant Measures on \(\mathbb{R}\)

    • Alexander Kharazishvili
    Pages 183-192

  15. Back Matter

    Pages 193-253
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Keywords

About this book

This monograph gives the reader an up-to-date account of the fine properties of real-valued functions and measures. The unifying theme of the book is the notion of nonmeasurability, from which one gets a full understanding of the structure of the subsets of the real line and the maps between them. The material covered in this book will be of interest to a wide audience of mathematicians, particularly to those working in the realm of real analysis, general topology, and probability theory. Set theorists interested in the foundations of real analysis will find a detailed discussion about the relationship between certain properties of the real numbers and the ZFC axioms, Martin's axiom, and the continuum hypothesis.

Reviews

“This monograph deals with classical topics of real analysis and measure theory which show a number of interesting phenomena. … This makes the presented material useful and inspiring. … Every chapter is finished with a solid portion of exercises … of various difficulty. More advanced exercises are enriched with hints and comments.” (Marek Balcerzak, Mathematical Reviews, June, 2023)




“The text is mostly self-contained and at the end of each chapter are exercises providing additional information to the presented topic. It makes the book accessible to graduate and post-graduate students.” (Jaroslav Tišer, zbMATH 1504.26003, 2023)

Authors and Affiliations

  • Andrea Razmadze Mathematical Institute, Tbilisi State University, Tbilisi, Georgia

    Alexander Kharazishvili

About the author

Alexander Kharazishvili is a Professor of Mathematics at I. Chavachavadze Tibilisi State University in Georgia. An expert in classical Real Analysis in the tradition of the Lusin school, he is the author of the well known monograph Strange Functions in Real Analysis.

Bibliographic Information

  • Book Title: Notes on Real Analysis and Measure Theory

  • Book Subtitle: Fine Properties of Real Sets and Functions

  • Authors: Alexander Kharazishvili

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-031-17033-1

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022

  • Hardcover ISBN: 978-3-031-17032-4Published: 24 September 2022

  • Softcover ISBN: 978-3-031-17035-5Published: 25 September 2023

  • eBook ISBN: 978-3-031-17033-1Published: 23 September 2022

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 1

  • Number of Pages: XI, 253

  • Topics: Mathematics, general

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