Aspects of Integration

Ronald B. Guenther is an emeritus professor in the Department of Mathematics at Oregon State University. His career began at the Marathon Oil Company where he served as an advanced research mathematician at its Denver Research Center. Most of his career was spent at Oregon State University, with visiting professorships at the Universities of Hamburg and Augsburg, and appointments at research laboratories in the United States and Canada, and at the Hahn-Meitner and Weierstrass Institutes in Berlin. His research interests include mathematically modeling deterministic systems and the ordinary and partial differential equations that arise from these models. John W. Lee is an Emeritus Professor in the Department of Mathematics at Oregon State University, where he spent his entire career with sabbatical leaves at Colorado State University, Montana State University, and many visits as a guest of Andrzej Granas at the University of Montreal. His research interests include topological methods use to study nonlinear differential and integral equations, oscillatory properties of problems of Sturm-Liouville type and related approximation theory, and various aspects of functional analysis and real analysis, especially measure and integration.

I. A Novel Approach to Riemann Integration. 1. Preliminaries. 1.1. Sums of Powers of Positive Integers. 1.2. Bernstein Polynomials. 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. 3. Extension to Higher Dimensions. 3.1. Method of Exhaustion. 3.2. Bernstein Polynomials in 2 Dimensions. 3.3. Integral of a Continuous Function. 4. Extension to the Lebesgue Integral. 4.1. Convergence and Cauchy Sequences. 4.2. Completion of the Rational Numbers. 4.3. Completion of C in the 1-norm. II. Lebesgue Integration. 5. Riesz-Nagy Approach. 5.1. Null Sets and Sets of Measure Zero. 5.2. Lemmas A and B. 5.3. The Class C1. 5.4. The Class C2. 5.5. Convergence Theorems. 5.6. Completeness. 5.7. The C2-Integral is the Lebesgue Integral. 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. A. Dinis Lemma. B. Semicontinuity. C. Completion of a Normed Linear Space. Bibliography. Index.