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.
Arvustused
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 Tier, zbMATH 1504.26003, 2023)
Preface.-
1. Real-Valued Semicontinuous Functions.-
2. The Oscillations of Real-Valued Functions.-
3. Monotone and Continuous Restrictions of Real-Valued Functions.-
4. Bijective Continuous Images of Absolute Null Sets.-
5. Projective Absolutely Nonmeasurable Functions.-
6. Borel Isomorphisms of Analytic Sets.-
7. Iterated Integrals of Real-Valued Functions of Two Real Variables.-
8. The Steinhaus Property, Ergocidity, and Density Points.-
9. Measurability Properties of H-Selectors and Partial H-Selectors.-
10. A Decomposition of an Uncountable Solvable Group into Three Negligible Sets.-
11. Negligible Sets Versus Absolutely Nonmeasurable Sets.-
12. Measurability Properties of Mazurkiewicz Sets.-
13. Extensions of Invariant Measures on R.- A. A Characterization of Uncountable Sets in Terms of their Self-Mappings.- B. Some Applications of Peano Type Functions.- C. Almost Rigid Mathematical Structures.- D. Some Unsolved Problems in Measure Theory.- Bibliography.- Index.
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.
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