Measure and Integration : A First Course

This concise text is intended as an introductory course in measure and integration . It covers essentials of the subject, providing ample motivation for new concepts and theorems in the form of discussion and remarks, and with many worked-out examples. The novelty of Measure and Integration: A First Course is in its style of exposition of the standard material in a student-friendly manner. New concepts are introduced progressively from less abstract to more abstract so that the subject is felt on solid footing. The book starts with a review of Riemann integration as a motivation for the necessity of introducing the concepts of measure and integration in a general setting. Then the text slowly evolves from the concept of an outer measure of subsets of the set of real line to the concept of Lebesgue measurable sets and Lebesgue measure, and then to the concept of a measure, measurable function, and integration in a more general setting. Again, integration is first introduced with non-negative functions, and then progressively with real and complex-valued functions. A chapter on Fourier transform is introduced only to make the reader realize the importance of the subject to another area of analysis that is essential for the study of advanced courses on partial differential equations. Key Features Numerous examples are worked out in detail. Lebesgue measurability is introduced only after convincing the reader of its necessity. Integrals of a non-negative measurable function is defined after motivating its existence as limits of integrals of simple measurable functions. Several inquisitive questions and important conclusions are displayed prominently. A good number of problems with liberal hints is provided at the end of each chapter. The book is so designed that it can be used as a text for a one-semester course during the first year of a master's program in mathematics or at the senior undergraduate level. About the Author M. Thamban Nair is a professor of mathematics at the Indian Institute of Technology Madras, Chennai, India. He was a post-doctoral fellow at the University of Grenoble, France through a French government scholarship, and also held visiting positions at Australian National University, Canberra, University of Kaiserslautern, Germany, University of St-Etienne, France, and Sun Yat-sen University, Guangzhou, China. The broad area of Prof. Nair’s research is in functional analysis and operator equations, more specifically, in the operator theoretic aspects of inverse and ill-posed problems. Prof. Nair has published more than 70 research papers in nationally and internationally reputed journals in the areas of spectral approximations, operator equations, and inverse and ill-posed problems. He is also the author of three books: Functional Analysis: A First Course (PHI-Learning, New Delhi), Linear Operator Equations: Approximation and Regularization (World Scientific, Singapore), and Calculus of One Variable (Ane Books Pvt. Ltd, New Delhi), and he is also co-author of Linear Algebra (Springer, New York). Cover 1 Half Title 2 Title Page 4 Copyright Page 5 Contents 6 Preface 8 Author 10 Note to the Reader 12 1. Review of Riemann Integral 14 1.1 Definition and Some Characterizations 14 1.2 Advantages and Some Disadvantages 21 1.3 Notations and Conventions 24 2. Lebesgue Measure 28 2.1 Lebesgue Outer Measure 28 2.2 Lebesgue Measurable Sets 36 2.3 Problems 46 3. Measure and Measurable Functions 50 3.1 Measure on an Arbitrary σ-Algebra 50 3.1.1 Lebesgue measure on Rk 54 3.1.2 Generated σ-algebra and Borel σ-algebra 55 3.1.3 Restrictions of σ-algebras and measures 58 3.1.4 Complete measure space and the completion 61 3.1.5 General outer measure and induced measure 63 3.2 Some Properties of Measures 65 3.3 Measurable Functions 69 3.3.1 Probability space and probability distribution 73 3.3.2 Further properties of measurable functions 74 3.3.3 Sequences and limits of measurable functions 77 3.3.4 Almost everywhere properties 79 3.4 Simple Measurable Functions 84 3.4.1 Measurability using simple measurable functions 88 3.4.2 Incompleteness of Borel σ-algebra 88 3.5 Problems 89 4. Integral of Positive Measurable Functions 94 4.1 Integral of Simple Measurable Functions 94 4.2 Integral of Positive Measurable Functions 101 4.2.1 Riemann integral as Lebesgue integral 108 4.2.2 Monotone convergence theorem (MCT) 110 4.2.3 Radon-Nikodym theorem 116 4.2.4 Conditional expectation 117 4.3 Appendix: Proof of the Radon-Nikodym Theorem 118 4.4 Problems 124 5. Integral of Complex Measurable Functions 126 5.1 Integrability and Some Properties 126 5.1.1 Riemann integral as Lebesgue integral 132 5.1.2 Dominated convergence theorem (DCT) 134 5.2 Lp Spaces 140 5.2.1 Hölder's and Minkowski's inequalities 142 5.2.2 Completeness of Lp (μ) 146 5.2.3 Denseness of Cc(Ω) in Lp(Ω) for 1 ≤ p < ∞ 151 5.3 Fundamental Theorems 153 5.3.1 Indefinite integral and its derivative 153 5.3.2 Fundamental theorems of Lebesgue integration 154 5.4 Appendix 164 5.5 Problems 170 6. Integration on Product Spaces 174 6.1 Motivation 174 6.2 Product σ-algebra and Product Measure 175 6.3 Fubini's Theorem 182 6.4 Counter Examples 185 6.4.1 σ-finiteness condition cannot be dropped 185 6.4.2 Product of complete measures need not be complete 186 6.5 Problems 186 7. Fourier Transform 190 7.1 Fourier Transform on L1 (R) 190 7.1.1 Definition and some basic properties 190 7.1.2 Fourier transform as a linear operator 198 7.1.3 Fourier inversion theorem 200 7.2 Fourier-Plancherel Transform 204 7.3 Problems 210 Bibliography 212 Index 214 The concepts from the theory of measure and integration are vital to any advanced course in analysis specifically in the applications of functional analysis to other areas such as harmonic analysis, partial differential equations, and integral equations. The book is meant for a one-semester course for the graduates of mathematics.