Aspects of Integration: Novel Approaches to the Riemann and Lebesgue Integrals Ronald

Aspects of Integration: Novel Approaches to the Riemann and Lebesgue Integrals is comprised of two parts. The first part is devoted to the Riemann integral, and provides not only a novel approach, but also includes several neat examples that are rarely found in other treatments of Riemann integration. Historical remarks trace the development of integration from the method of exhaustion of Eudoxus and Archimedes, used to evaluate areas related to circles and parabolas, to Riemann’s careful definition of the definite integral, which is a powerful expansion of the method of exhaustion and makes it clear what a definite integral really is. The second part follows the approach of Riesz and Nagy in which the Lebesgue integral is developed without the need for any measure theory. Our approach is novel in part because it uses integrals of continuous functions rather than integrals of step functions as its starting point. This is natural because Riemann integrals of continuous functions occur much more frequently than do integrals of step functions as a precursor to Lebesgue integration. In addition, the approach used here is natural because step functions play no role in the novel development of the Riemann integral in the first part of the book. Our presentation of the Riesz-Nagy approach is significantly more accessible, especially in its discussion of the two key lemmas upon which the approach critically depends, and is more concise than other treatments. Features Presents novel approaches designed to be more accessible than classical presentations. A welcome alternative approach to the Riemann integral in undergraduate analysis courses. Makes the Lebesgue integral accessible to upper division undergraduate students. How completion of the Riemann integral leads to the Lebesgue integral. Contains a number of historical insights. Gives added perspective to researchers and postgraduates interested in the Riemann and Lebesgue integrals. Cover Half Title Series Page Title Page Copyright Page Contents Contributors Preface To the Reader Acknowledgement Part I: A Novel Approach to Riemann Integration CHAPTER 1. Preliminaries 1.1 Sums of Powers of Positive Integers 1.2 Bernstein Polynomials CHAPTER 2. The Riemann Integral 2.1 Method of Exhaustion 2.2 Integral of a Continuous Function 2.3 Foundational Theorems of Integral Calculus 2.4 Integration by Substitution CHAPTER 3. Extension to Higher Dimensions 3.1 Method of Exhaustion 3.2 Bernstein Polynomials in 2 Dimensions 3.3 Integral of a Continuous Function Integrals Over More General Domains Iterated Integration CHAPTER 4. Extension to the Lebesgue Integral 4.1 Convergence and Cauchy Sequences 4.2 Completion of the Rational Numbers Conundrum Calculus 4.3 Completion of C in the 1-norm Part II: Lebesgue Integration Chapter 5. The Riesz-Nagy Approach to the Lebesgue Integral 5.1 Null Sets and Sets of Measure Zero 5.2 Lemma's A and B 5.3 The Class C1(I) of Riesz and Nagy 5.4 The Class C2 of Riesz and Nagy 5.5 Convergence Theorems 5.6 Completeness 5.7 The C2-Integral is the Lebesgue Integral CHAPTER 6. Comparing Integrals 6.1 Properly Integrable Functions 6.2 Characterization of the Riemann Integral 6.3 Riemann vs. Lebesgue Integrals 6.4 The Novel Approach Appendix A. Dini's Lemma Appendix B. Semicontinuity Appendix C. Completion of a Normed Linear Space References Index This book is comprised of two parts. The first part is devoted to the Riemann integral, and provides a novel approach, that are rarely found in other treatments of Riemann integration. The second part follows the approach of Riesz and Nagy in which the Lebesgue integral is developed without the need for any measure theory.