Kharazishvili A. Notes on Real Analysis and Measure Theory 2022

Textbook in PDF format

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
The Oscillations of Real-Valued Functions
Monotone and Continuous Restrictions of Real-Valued Functions
Bijective Continuous Images of Absolute Null Sets
Projective Absolutely Nonmeasurable Functions
Borel Isomorphisms of Analytic Sets
Iterated Integrals of Real-Valued Functions of Two Real Variables
The Steinhaus Property, Ergodicity, and Density Points
Measurability Properties of H-Selectors and Partial H-Selectors
A Decomposition of an Uncountable Solvable Group into Three Negligible Sets
Negligible Sets Versus Absolutely Nonmeasurable Sets
Measurability Properties of Mazurkiewicz Sets
Extensions of Invariant Measures on R