Notes on Real Analysis and Measure Theory

Description

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. Real-Valued Semicontinuous Functions 1-19 The Oscillations of Real-Valued Functions 21-35 Monotone and Continuous Restrictions of Real-Valued Functions 37-49 Bijective Continuous Images of Absolute Null Sets 51-64 Projective Absolutely Nonmeasurable Functions 65-75 Borel Isomorphisms of Analytic Sets 77-92 Iterated Integrals of Real-Valued Functions of Two Real Variables 93-103 The Steinhaus Property, Ergodicity, and Density Points 105-121 Measurability Properties of H-Selectors and Partial H-Selectors 123-141 A Decomposition of an Uncountable Solvable Group into Three Negligible Sets 143-157 Negligible Sets Versus Absolutely Nonmeasurable Sets 159-168 Measurability Properties of Mazurkiewicz Sets 169-182 Extensions of Invariant Measures on R 183-192