TOPICS IN MEASURE THEORY AND REAL ANALYSIS

1. Front Matter

   Pages i-xiv

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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

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

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

  Alexander B. Kharazishvili

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