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TOPICS IN MEASURE THEORY AND REAL ANALYSIS

  • Book
  • © 2009

Overview

Authors:
  1. Alexander B. Kharazishvili

    1. A. Razmadze Mathematical Institute, Tbilisi, Republic of Georgia

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

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Table of contents (20 chapters)

  1. Front Matter

    Pages i-xiv
    Download chapter PDF

  2. The problem of extending partial functions

    • Alexander B. Kharazishvili
    Pages 1-18

  3. Some aspects of the measure extension problem

    • Alexander B. Kharazishvili
    Pages 19-37

  4. Invariant measures

    • Alexander B. Kharazishvili
    Pages 39-62

  5. Quasi-invariant measures

    • Alexander B. Kharazishvili
    Pages 63-78

  6. Measurability properties of real-valued functions

    • Alexander B. Kharazishvili
    Pages 79-96

  7. Some properties of step-functions connected with extensions of measures

    • Alexander B. Kharazishvili
    Pages 97-109

  8. Almost measurable real-valued functions

    • Alexander B. Kharazishvili
    Pages 111-124

  9. Several facts from general topology

    • Alexander B. Kharazishvili
    Pages 125-143

  10. Weakly metrically transitive measures and nonmeasurable sets

    • Alexander B. Kharazishvili
    Pages 145-158

  11. Nonmeasurable subgroups of uncountable solvable groups

    • Alexander B. Kharazishvili
    Pages 159-175

  12. Algebraic sums of measure zero sets

    • Alexander B. Kharazishvili
    Pages 177-198

  13. The absolute nonmeasurability of Minkowski’s sum of certain universal measure zero sets

    • Alexander B. Kharazishvili
    Pages 199-213

  14. Absolutely nonmeasurable additive Sierpiński-Zygmund functions

    • Alexander B. Kharazishvili
    Pages 215-226

  15. Relatively measurable Sierpiński-Zygmund functions

    • Alexander B. Kharazishvili
    Pages 227-239

  16. A nonseparable extension of the Lebesgue measure without new null-sets

    • Alexander B. Kharazishvili
    Pages 241-256

  17. Metrical transitivity and nonseparable extensions of invariant measures

    • Alexander B. Kharazishvili
    Pages 257-268

  18. Nonseparable left invariant measures on uncountable solvable groups

    • Alexander B. Kharazishvili
    Pages 269-280

  19. Universally measurable additive functionals

    • Alexander B. Kharazishvili
    Pages 281-295

  20. Some subsets of the Euclidean plane

    • Alexander B. Kharazishvili
    Pages 297-312
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Keywords

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.

Authors and Affiliations

  • A. Razmadze Mathematical Institute, Tbilisi, Republic of Georgia

    Alexander B. Kharazishvili

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