# Set Theoretical Aspects of Real Analysis

The present book is devoted to a circle of questions in real analysis and classical measure theory, which are of a somewhat set-theoretic flavor. We are focused on certain logical and set-theoretical aspects of real analysis, ...

Author: Alexander B. Kharazishvili

Publisher: CRC Press

ISBN: 9781482242027

Category: Mathematics

Page: 456

View: 800

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 b
Categories: Mathematics

# Set Theoretical Aspects of Real Analysis

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.

Author: Alexander B. Kharazishvili

Publisher: CRC Press

ISBN: 9781482242010

Category: Mathematics

Page: 456

View: 731

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.
Categories: Mathematics

# Applications of Point Set Theory in Real Analysis

c) various constructions of Lebesgue nonmeasurable sets and of sets without the Baire property; some connections of ... with the general theory of commutative groups are indicated; d) various singular objects in mathematical analysis ...

Author: A.B. Kharazishvili

Publisher: Springer Science & Business Media

ISBN: 9789401707503

Category: Mathematics

Page: 240

View: 302

This book is devoted to some results from the classical Point Set Theory and their applications to certain problems in mathematical analysis of the real line. Notice that various topics from this theory are presented in several books and surveys. From among the most important works devoted to Point Set Theory, let us first of all mention the excellent book by Oxtoby [83] in which a deep analogy between measure and category is discussed in detail. Further, an interesting general approach to problems concerning measure and category is developed in the well-known monograph by Morgan [79] where a fundamental concept of a category base is introduced and investigated. We also wish to mention that the monograph by Cichon, W«;glorz and the author [19] has recently been published. In that book, certain classes of subsets of the real line are studied and various cardinal valued functions (characteristics) closely connected with those classes are investigated. Obviously, the IT-ideal of all Lebesgue measure zero subsets of the real line and the IT-ideal of all first category subsets of the same line are extensively studied in [19], and several relatively new results concerning this topic are presented. Finally, it is reasonable to notice here that some special sets of points, the so-called singular spaces, are considered in the classi
Categories: Mathematics

# Real Analysis

This book would be useful as text for undergraduate students of all Indian universities and engineering institutes, including the Indian Institutes of Technology. Real Analysis is a CORE subject in mathematics at the college level.

Author: S. Nanda

Publisher: Allied Publishers

ISBN: 9788177640625

Category:

Page: 288

View: 686

This book would be useful as text for undergraduate students of all Indian universities and engineering institutes, including the Indian Institutes of Technology. Real Analysis is a CORE subject in mathematics at the college level. The prerequisite for this course is Higher Secondary level mathematics including calculus. The authors have, however, included a preliminary chapter on Set Theory to make the book as self contained as possible. In addition to discussing the “basics” of a first course, the book also contains a large number of examples to aid better student understanding of the subject.
Categories:

# TOPICS IN MEASURE THEORY AND REAL ANALYSIS

This book is concerned with questions of classical measure theory and related topics of real analysis. ... In addition, we touch upon deep set-theoretical aspects of the topics discussed in the book; consequently, set-theorists may ...

Author: Alexander Kharazishvili

Publisher: Springer Science & Business Media

ISBN: 9789491216367

Category: Mathematics

Page: 461

View: 629

This book highlights various topics on measure theory and vividly demonstrates that the different questions of this theory are closely connected with the central measure extension problem. Several important aspects of the measure extension problem are considered separately: set-theoretical, topological and algebraic. Also, various combinations (e.g., algebraic-topological) of these aspects are discussed by stressing their specific features. Several new methods are presented for solving the above mentioned problem in concrete situations. In particular, the following new results are obtained: the measure extension problem is completely solved for invariant or quasi-invariant measures on solvable uncountable groups; non-separable extensions of invariant measures are constructed by using their ergodic components; absolutely non-measurable additive functionals are constructed for certain classes of measures; the structure of algebraic sums of measure zero sets is investigated. The material presented in this book is essentially self-contained and is oriented towards a wide audience of mathematicians (including postgraduate students). New results and facts given in the book are based on (or closely connected with) traditional topics of set theory, measure theory and general topology such as: infinite combinatorics, Martin's Axiom and the Continuum Hypothesis, Luzin and Sierpinski sets, universal measure zero sets, theorems on the existence of measurable selectors, regularity properties of Borel measures on metric spaces, and so on. Essential information on these topics is also included in the text (primarily, in the form of Appendixes or Exercises), which enables potential readers to understand the proofs and follow the constructions in full details. This not only allows the book to be used as a monograph but also as a course of lectures for students whose interests lie in set theory, real analysis, measure theory and general topology.
Categories: Mathematics

# Strange Functions in Real Analysis Third Edition

than the Continuum Hypothesis but rather helpful in various constructions of set theory, topology, measure theory, and real analysis (cf. [18], [146]). Then we briefly present some basic concepts of general topology and classical ...

Author: Alexander Kharazishvili

Publisher: CRC Press

ISBN: 9781351650519

Category: Mathematics

Page: 426

View: 820

Strange Functions in Real Analysis, Third Edition differs from the previous editions in that it includes five new chapters as well as two appendices. More importantly, the entire text has been revised and contains more detailed explanations of the presented material. In doing so, the book explores a number of important examples and constructions of pathological functions. After introducing basic concepts, the author begins with Cantor and Peano-type functions, then moves effortlessly to functions whose constructions require what is essentially non-effective methods. These include functions without the Baire property, functions associated with a Hamel basis of the real line and Sierpinski-Zygmund functions that are discontinuous on each subset of the real line having the cardinality continuum. Finally, the author considers examples of functions whose existence cannot be established without the help of additional set-theoretical axioms. On the whole, the book is devoted to strange functions (and point sets) in real analysis and their applications.
Categories: Mathematics

# Transformation Groups and Invariant Measures

[131] Wilczynski W, A generalization of the density topology, Real Analysis Exchange, vol. 8, no. 1, 1982 - 1983, pp. ... [136] Zakrzewski P, The uniqueness of Haar measure and set theory, Colloq. Math, vol. 74, no. 1, 1997, pp.

Author: A B Kharazishvili

Publisher: World Scientific

ISBN: 9789814518222

Category: Mathematics

Page: 268

View: 171

This book is devoted to some topics of the general theory of invariant and quasi-invariant measures. Such measures are usually defined on various σ-algebras of subsets of spaces equipped with transformation groups, and there are close relationships between purely algebraic properties of these groups and the corresponding properties of invariant (quasi-invariant) measures. The main goal of the book is to investigate several aspects of those relationships (primarily from the set-theoretical point of view). Also of interest are the properties of some natural classes of sets, important from the viewpoint of the theory of invariant (quasi-invariant) measures. Contents:Some Properties of Transformation GroupsQuasiinvariant and Invariant MeasuresSome Examples and ConstructionsNonmeasurable Sets with Respect to Quasiinvariant and Invariant MeasuresSmall Sets with Respect to Quasiinvariant MeasuresAlmost Invariant SetsSome Invariant σ-Ideals and σ-AlgebrasDensity Points and Invariant Extensions of Lebesgue MeasureThe Uniqueness of Lebesgue and Borel MeasuresQuasiinvariant Borel Measures on Standard Groups Readership: Pure mathematicians. Keywords:Transformation Group;Invariant Measure;Quasi-Invariant Measure;Absolutely Negligible Set;Absolutely Nonmeasurable Set;Extension of Measure;Haar Measure;Lebesgue Measure;Uniqueness Property for Measures;Steinhaus Property;Metrical Transitivity;Standard Group;Measurable Cardinal
Categories: Mathematics

# Introductory Real Analysis

Comprehensive, elementary introduction to real and functional analysis covers basic concepts and introductory principles in set theory, metric spaces, topological and linear spaces, linear functionals and linear operators, more. 1970 ...

Author: A. N. Kolmogorov

Publisher: Courier Corporation

ISBN: 9780486612263

Category: Mathematics

Page: 403

View: 131

Comprehensive, elementary introduction to real and functional analysis covers basic concepts and introductory principles in set theory, metric spaces, topological and linear spaces, linear functionals and linear operators, more. 1970 edition.
Categories: Mathematics

These works present a comprehensive treatment with a global view of the subject, emphasizing the connections between real analysis and other branches of mathematics.

Author: Anthony W. Knapp

Publisher: Springer Science & Business Media

ISBN: 9780817644420

Category: Mathematics

Page: 466

View: 811

* Presents a comprehensive treatment with a global view of the subject * Rich in examples, problems with hints, and solutions, the book makes a welcome addition to the library of every mathematician
Categories: Mathematics

# An Introduction to Real Analysis

Each chapter contains exercises meant to facilitate understanding of the subject matter. This book is intended for students in colleges of education and others with similar needs.

Author: Derek G. Ball

Publisher: Elsevier

ISBN: 9781483158969

Category: Mathematics

Page: 324

View: 618

An Introduction to Real Analysis presents the concepts of real analysis and highlights the problems which necessitate the introduction of these concepts. Topics range from sets, relations, and functions to numbers, sequences, series, derivatives, and the Riemann integral. This volume begins with an introduction to some of the problems which are met in the use of numbers for measuring, and which provide motivation for the creation of real analysis. Attention then turns to real numbers that are built up from natural numbers, with emphasis on integers, rationals, and irrationals. The chapters that follow explore the conditions under which sequences have limits and derive the limits of many important sequences, along with functions of a real variable, Rolle's theorem and the nature of the derivative, and the theory of infinite series and how the concepts may be applied to decimal representation. The book also discusses some important functions and expansions before concluding with a chapter on the Riemann integral and the problem of area and its measurement. Throughout the text the stress has been upon concepts and interesting results rather than upon techniques. Each chapter contains exercises meant to facilitate understanding of the subject matter. This book is intended for students in colleges of education and others with similar needs.
Categories: Mathematics