Set Theoretical Aspects of Real Analysis (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

This book addresses a number of questions in real analysis and classical measure theory that are of a set-theoretic flavor. Accessible to graduate students, the beginning of the book presents introductory topics on real analysis and Lebesque measure theory. These topics highlight the boundary between fundamental concepts of measurability and non-measurability for point sets and functions. The remainder of the book deals with more specialized material on set-theoretical real analysis. Problems are included at the end of each chapter. MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict MuPDF error: syntax error: invalid key in dict Front Cover 1 Table of Contents 8 Preface 10 Chapter 1: ZF Theory and Some Point Sets on the Real Line 24 Chapter 2: Countable Versions of AC and Real Analysis 44 Chapter 3: Uncountable Versions of AC and Lebesgue Nonmeasurable Sets 58 Chapter 4: The Continuum Hypothesis and Lebesgue Nonmeasurable Sets 76 Chapter 5: Measurability Properties of Sets and Functions 90 Chapter 6: Radon Measures and Nonmeasurable Sets 110 Chapter 7: Real-Valued Step Functions with Strange Measurability Properties 130 Chapter 7: A Partition of the Real Line Into Continuum Many Thick Subsets 146 Chapter 9: Measurability Properties of Vitali Sets 160 Chapter 10: A Relationship Between the Measurability and Continuity of Real-Valued Functions 174 Chapter 11: A Relationship Between Absolutely Nonmeasurable Functions and Sierpiński–Zygmund Type Functions 190 Chapter 12: Sums of Absolutely Nonmeasurable Injective Functions 204 Chapter 13: A Large Group of Absolutely Nonmeasurable Additive Functions 218 Chapter 14: Additive Properties of Certain Classes of Pathological Functions 232 Chapter 15: Absolutely Nonmeasurable Homomorphisms of Commutative Groups 248 Chapter 16: Measurable and Nonmeasurable Sets With Homogeneous Sections 262 Chapter 17: A Combinatorial Problem on Translation Invariant Extensions of the Lebesgue Measure 276 Chapter 18: Countable Almost Invariant Partitions of G-Spaces 292 Chapter 19: Nonmeasurable Unions of Measure Zero Sections of Plane Sets 310 Chapter 20: Measurability Properties of Well-Orderings 322 Appendix 1: The Axioms of Set Theory 340 Appendix 2: The Axiom of Choice and Generalized Continuum Hypothesis 364 Appendix 3: Martin’s Axiom and its consequences in real analysis 378 Appendix 4: ω1-dense subsets of the real line 394 Appendix 5: The Beginnings of Descriptive Set Theory 404 Bibliography 434 Back Cover 452 Set Theoretical Aspects of Real Analysis is built around a number of questions in real analysis and classical measure theory, which are of a set theoretic flavor. Accessible to graduate students, and researchers the beginning of the book presents introductory topics on real analysis and Lebesgue measure theory. These topics highlight the boundary between fundamental concepts of measurability and nonmeasurability for point sets and functions. The remainder of the book deals with more specialized material on set theoretical real analysis. The book focuses on certain logical and set theoretical aspects of real analysis. It is expected that the first eleven chapters can be used in a course on Lebesque measure theory that highlights the fundamental concepts of measurability and non-measurability for point sets and functions. Provided in the book are problems of varying difficulty that range from simple observations to advanced results. Relatively difficult exercises are marked by asterisks and hints are included with additional explanation. Five appendices are included to supply additional background information that can be read alongside, before, or after the chapters. Dealing with classical concepts, the book highlights material not often found in analysis courses. It lays out, in a logical, systematic manner, the foundations of set theory providing a readable treatment accessible to graduate students and researchers. Features: --Adopts a new ideology based on a classification of real valued functions (point sets) according to their measurability properties. --Discusses the logical relationships between classical constructions of Lebesgue nonmeasureable sets. --Covers the structure of absolutely non-measurable real valued functions. --Presents a characteristic of all commutative groups that admit absolutely non-measurable homomorphisms into the real line. --Explores interesting and unexpected connections between combinatorics and translation-invariant extensions of the Lebesgue measure. Alexander B. Kharazishvili is Full professor at the A. Razmadze Mathematical Institute of I. Chavchavadze State University, Tblisi, Georgia. Publisher's note 1. ZF theory and some point sets on the real line -- -- 2. Countable versions of AC and real analysis -- -- 3. Uncountable versions of AC and Lebesgue nonmeasurable sets -- -- 4. The Continuum Hypothesis and Lebesgue nonmeasurable sets -- -- 5. Measurability properties of sets and functions -- -- 6. Radon measures and nonmeasurable sets -- -- 7. Real-valued step functions with strange measurability properties -- -- 8. A partition of the real line into continuum many thick subsets -- -- 9. Measurability properties of Vitali sets -- -- 10. A relationship between the measurability and continuity of real-valued functions -- -- 11. A relationship between absolutely nonmeasurable functions and Sierpiński-Zygmund type functions -- -- 12. Sums of absolutely nonmeasurable injective functions -- -- 13. A large group of absolutely nonmeasurable additive functions -- -- 14. Additive properties of certain classes of pathological functions -- -- 15. Absolutely nonmeasurable homomorphisms of commutative groups -- -- 16. Measurable and nonmeasurable sets with homogeneous sections -- -- 17. A combinatorial problem on translation invariant extensions of the Lebesgue measure -- -- 18. Countable almost invariant partitions of G-spaces -- -- 19. Nonmeasurable unions of measure zero sections of plane sets -- -- 20. Measurability properties of well-orderings.