Notes on Real Analysis and Measure Theory

Fine Properties of Real Sets and Functions

Häftad, Engelska, 2023

Av Alexander Kharazishvili

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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.

Produktinformation

Utgivningsdatum2023-09-25
Mått155 x 235 x 15 mm
Vikt411 g
FormatHäftad
SpråkEngelska
SerieSpringer Monographs in Mathematics
Antal sidor253
FörlagSpringer International Publishing AG
ISBN9783031170355

Tillhör följande kategorier

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.

“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 Tišer, zbMATH 1504.26003, 2023)