Problems in Mathematical Analysis III : Integration

The best way to penetrate the subtleties of the theory of integration is by solving problems. This book, like its two predecessors, is a wonderful source of interesting and challenging problems. As a resource, it is unequaled. It offers a much richer selection than is found in any current textbook. Moreover, the book includes a complete set of solutions. This is the third volume of ""Problems in Mathematical Analysis"". The topic here is integration for real functions of one real variable.The first chapter is devoted to the Riemann and the Riemann-Stieltjes integrals. Chapter 2 deals with Lebesgue measure and integration. The authors include some famous, and some not so famous, inequalities related to Riemann integration. Many of the problems for Lebesgue integration concern convergence theorems and the interchange of limits and integrals. The book closes with a section on Fourier series, with a concentration on Fourier coefficients of functions from particular classes and on basic theorems for convergence of Fourier series.The book is mainly geared toward students studying the basic principles of analysis. However, given its selection of problems, organization, and level, it would be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. It is also suitable for self-study. The presentation of the material is designed to help student comprehension, to encourage them to ask their own questions, and to start research. The collection of problems will also help teachers who wish to incorporate problems into their lectures. The problems are grouped into sections according to the methods of solution. Solutions for the problems are provided. ""Problems in Mathematical Analysis I and II"" are available as Volumes 4 and 12 in the AMS series, ""Student Mathematical Library"" We learn by doing. We learn mathematics by doing problems. This book is the first volume of a series of books of problems in mathematical analysis. It is mainly intended for students studying the basic principles of analysis. However, given its organization, level, and selection of problems, it would also be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. The volume is also suitable for self-study. Each section of the book begins with relatively simple exercises, yet may also contain quite challenging problems. Very often several consecutive exercises are concerned with different aspects of one mathematical problem or theorem. This presentation of material is designed to help student comprehension and to encourage them to ask their own questions and to start research. The collection of problems in the book is also intended to help teachers who wish to incorporate the problems into lectures. Solutions for all the problems are provided. The book covers three real numbers, sequences, and series, and is divided into two exercises and/or problems, and solutions. Specific topics covered in this volume include the basic properties of real numbers, continued fractions, monotonic sequences, limits of sequences, Stolz's theorem, summation of series, tests for convergence, double series, arrangement of series, Cauchy product, and infinite products. Cover Title Copyright Contents Preface Part 1. Problems Chapter 1. The Riemann-Stieltjes Integral §1.1. Properties of the Riemann-Stieltjes Integral §1.2. Functions of Bounded Variation §1.3. Further Properties of the Riemann-Stieltjes Integral §1.4. Proper Integrals §1.5. Improper Integrals §1.6. Integral Inequalities §1.7. Jordan Measure Chapter 2. The Lebesgue Integral §2.1. Lebesgue Measure on the Real Line §2.2. Lebesgue Measurable Functions §2.3. Lebesgue Integration §2.4. Absolute Continuity, Differentiation and Integration §2.5. Fourier Series Part 2. Solutions Chapter 1. The Riemann-Stieltjes Integral §1.1. Properties of the Riemann-Stieltjes Integral §1.2. Functions of Bounded Variation §1.3. Further Properties of the Riemann-Stieltjes Integral §1.4. Proper Integrals §1.5. Improper Integrals §1.6. Integral Inequalities §1.7. Jordan Measure Chapter 2. The Lebesgue Integral §2.1. Lebesgue Measure on the Real Line §2.2. Lebesgue Measurable Functions §2.3. Lebesgue Integration §2.4. Absolute Continuity, Differentiation and Integration §2.5. Fourier Series Bibliography -Books Index A B C D E F H I J L M O P R S T U V W Y Back Cover We Learn By Doing. We Learn Mathematics By Doing Problems. This Is The Third Volume Of Problems In Mathematical Analysis. The Topic Here Is Integration For Real Functions Of One Real Variable. The First Chapter Is Devoted To The Riemann And The Riemann-stieltjes Integrals. Chapter 2 Deals With Lebesgue Measure And Integration. The Authors Include Some Famous, And Some Not So Famous, Integral Inequalities Related To Riemann Integration. Many Of The Problems For Lebesgue Integration Concern Convergence Theorems And The Interchange Of Limits And Integrals. The Book Closes With A Section On Fourier Series, With A Concentration On Fourier Coefficients Of Functions From Particular Classes And On Basic Theorems For Convergence Of Fourier Series. The Book Is Primarily Geared Toward Students In Analysis, As A Study Aid, For Problem-solving Seminars, Or For Tutorials. It Is Also An Excellent Resource For Instructors Who Wish To Incorporate Problems Into Their Lectures. Solutions For The Problems Are Provided In The Book.--résumé De L'éditeur. We learn by doing. We learn mathematics by doing problems. This book is the first volume of a series of books of problems in mathematical analysis. It is mainly intended for students studying the basic principles of analysis. However, given its organization, level, and selection of problems, it would also be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. The volume is also suitable for self-study. Each section of the book begins with relatively simple exercises, yet may also contain quite challenging problems. Very often a few consecutive exercises are concerned with different aspects of one mathematical problem or theorem. This presentation of material is designed to help student comprehension and to encourage them to ask their own questions and to start research. The collection of problems in the book is also intended to help teachers who wish to incorporate the problems into lectures. Solutions for all the problems are provided. A supplement to undergraduate and undergraduate textbooks in mathematical analysis. The third volume in the series focuses on the Riemann-Stieltjes integral and the Lebesgue integral for real functions of one real variable. The first half contains the problems, and the second the solutions and explanations. Annotation (c) Book News, Inc., Portland, OR (booknews.com) 1. Real Numbers, Sequences, And Series -- 2. Continuity And Differentiation -- 3. Integration. W.j. Kaczor, M.t. Nowak. Includes Bibliographical References.