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Applications of Point Set Theory in Real Analysis

  • Book
  • © 1998

Overview

Authors:
  1. A. B. Kharazishvili

    1. Institute of Applied Mathematics, Tbilisi State University, Tbilisi, USA

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Part of the book series: Mathematics and Its Applications (MAIA, volume 429)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-viii
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  2. Introduction: preliminary facts

    • A. B. Kharazishvili
    Pages 1-20

  3. Set-valued mappings

    • A. B. Kharazishvili
    Pages 21-38

  4. Nonmeasurable sets and sets without the Baire property

    • A. B. Kharazishvili
    Pages 39-54

  5. Three aspects of the measure extension problem

    • A. B. Kharazishvili
    Pages 55-76

  6. Some properties of σ-algebras and σ-ideals

    • A. B. Kharazishvili
    Pages 77-90

  7. Nonmeasurable subgroups of the real line

    • A. B. Kharazishvili
    Pages 91-100

  8. Additive properties of invariant σ-ideals on the real line

    • A. B. Kharazishvili
    Pages 101-110

  9. Translations of sets and functions

    • A. B. Kharazishvili
    Pages 111-122

  10. The Steinhaus property of invariant measures

    • A. B. Kharazishvili
    Pages 123-132

  11. Some applications of the property (N) of Luzin

    • A. B. Kharazishvili
    Pages 133-142

  12. The principle of condensation of singularities

    • A. B. Kharazishvili
    Pages 143-160

  13. The uniqueness of Lebesgue and Borel measures

    • A. B. Kharazishvili
    Pages 161-172

  14. Some subsets of spaces equipped with transformation groups

    • A. B. Kharazishvili
    Pages 173-184

  15. Sierpiński’s partition and its applications

    • A. B. Kharazishvili
    Pages 185-196

  16. Selectors associated with subgroups of the real line

    • A. B. Kharazishvili
    Pages 197-208

  17. Set theory and ordinary differential equations

    • A. B. Kharazishvili
    Pages 209-222

  18. Back Matter

    Pages 223-240
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Keywords

About this book

This book is devoted to some results from the classical Point Set Theory and their applications to certain problems in mathematical analysis of the real line. Notice that various topics from this theory are presented in several books and surveys. From among the most important works devoted to Point Set Theory, let us first of all mention the excellent book by Oxtoby [83] in which a deep analogy between measure and category is discussed in detail. Further, an interesting general approach to problems concerning measure and category is developed in the well-known monograph by Morgan [79] where a fundamental concept of a category base is introduced and investigated. We also wish to mention that the monograph by Cichon, W«;glorz and the author [19] has recently been published. In that book, certain classes of subsets of the real line are studied and various cardinal­ valued functions (characteristics) closely connected with those classes are investigated. Obviously, the IT-ideal of all Lebesgue measure zero subsets of the real line and the IT-ideal of all first category subsets of the same line are extensively studied in [19], and several relatively new results concerning this topic are presented. Finally, it is reasonable to notice here that some special sets of points, the so-called singular spaces, are considered in the classi

Reviews

.....`It is also recommended as a textbook for postgraduate students.'
European Mathematical Society News, December 1999

Authors and Affiliations

  • Institute of Applied Mathematics, Tbilisi State University, Tbilisi, USA

    A. B. Kharazishvili

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