Set Theoretical Aspects of Real Analysis: 1st Edition (Hardback) book cover

Set Theoretical Aspects of Real Analysis

1st Edition

By Alexander B. Kharazishvili

Chapman and Hall/CRC

456 pages

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Hardback: 9781482242010
pub: 2014-08-26
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Description

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.

Table of Contents

Preface

ZF theory and some point sets on the real line

Countable versions of AC and real analysis

Uncountable versions of AC and Lebesgue nonmeasurable sets

The Continuum Hypothesis and Lebesgue nonmeasurable sets

Measurability properties of sets and functions

Radon measures and nonmeasurable sets

Real-valued step functions with strange measurability properties

A partition of the real line into continuum many thick subsets

Measurability properties of Vitali sets

A relationship between the measurability and continuity of real-valued functions

A relationship between absolutely nonmeasurable functions and Sierpi´nski–Zygmund type functions

Sums of absolutely nonmeasurable injective functions

A large group of absolutely nonmeasurable additive functions

Additive properties of certain classes of pathological functions

Absolutely nonmeasurable homomorphisms of commutative groups

Measurable and nonmeasurable sets with homogeneous sections

A combinatorial problem on translation invariant extensions of the Lebesgue measure

Countable almost invariant partitions of G-spaces

Nonmeasurable unions of measure zero sections of plane sets

Measurability properties of well-orderings

Appendix 1: The axioms of set theory

Appendix 2: The Axiom of Choice and Generalized Continuum Hypothesis

Appendix 3: Martin’s Axiom and its consequences in real analysis

Appendix 4: !1-dense subsets of the real line

Appendix 5: The beginnings of descriptive set theory

Bibliography

Subject Index

About the Series

Chapman & Hall/CRC Monographs and Research Notes in Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT028000
MATHEMATICS / Set Theory
MAT037000
MATHEMATICS / Functional Analysis