Topics in Measure Theory and Real Analysis

  • Alexander B. Kharazishvili

Part of the Atlantis Studies in Mathematics book series (ATLANTISSM, volume 2)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Alexander B. Kharazishvili
    Pages 1-18
  3. Alexander B. Kharazishvili
    Pages 19-37
  4. Alexander B. Kharazishvili
    Pages 39-62
  5. Alexander B. Kharazishvili
    Pages 63-78
  6. Alexander B. Kharazishvili
    Pages 79-96
  7. Alexander B. Kharazishvili
    Pages 111-124
  8. Alexander B. Kharazishvili
    Pages 125-143
  9. Alexander B. Kharazishvili
    Pages 145-158
  10. Alexander B. Kharazishvili
    Pages 159-175
  11. Alexander B. Kharazishvili
    Pages 177-198
  12. Alexander B. Kharazishvili
    Pages 215-226
  13. Alexander B. Kharazishvili
    Pages 227-239
  14. Alexander B. Kharazishvili
    Pages 281-295
  15. Alexander B. Kharazishvili
    Pages 297-312
  16. Alexander B. Kharazishvili
    Pages 313-338
  17. Back Matter
    Pages 339-461

About this book


This book highlights various topics on measure theory and vividly demonstrates that the different questions of this theory are closely connected with the central measure extension problem. Several important aspects of the measure extension problem are considered separately: set-theoretical, topological and algebraic. Also, various combinations (e.g., algebraic-topological) of these aspects are discussed by stressing their specific features. Several new methods are presented for solving the above mentioned problem in concrete situations. In particular, the following new results are obtained: the measure extension problem is completely solved for invariant or quasi-invariant measures on solvable uncountable groups; non-separable extensions of invariant measures are constructed by using their ergodic components; absolutely non-measurable additive functionals are constructed for certain classes of measures; the structure of algebraic sums of measure zero sets is investigated. The material presented in this book is essentially self-contained and is oriented towards a wide audience of mathematicians (including postgraduate students). New results and facts given in the book are based on (or closely connected with) traditional topics of set theory, measure theory and general topology such as: infinite combinatorics, Martin's Axiom and the Continuum Hypothesis, Luzin and Sierpinski sets, universal measure zero sets, theorems on the existence of measurable selectors, regularity properties of Borel measures on metric spaces, and so on. Essential information on these topics is also included in the text (primarily, in the form of Appendixes or Exercises), which enables potential readers to understand the proofs and follow the constructions in full details. This not only allows the book to be used as a monograph but also as a course of lectures for students whose interests lie in set theory, real analysis, measure theory and general topology.


Finite Invariant Lebesgue measure algebra boundary element method functional information measure measure theory metric space proof set theory sets theorem topology

Authors and affiliations

  • Alexander B. Kharazishvili
    • 1
  1. 1.A. Razmadze Mathematical InstituteTbilisiRepublic of Georgia

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