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The Lebesgue-stieltjes Integral
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Mathematics students generally meet the Riemann integral early in their undergraduate studies, then at advanced undergraduate or graduate level they receive a course on measure and integration dealing with the Lebesgue theory. However, those whose interests lie more in the direction of applied mathematics will in all probability find themselves needing to use the Lebesgue or Lebesgue-Stieltjes Integral without having the necessary theoretical background. It is to such readers that this book is addressed. The authors aim to introduce the Lebesgue-Stieltjes integral on the real line in a natural way as an extension of the Riemann integral. They have tried to make the treatment as practical as possible. The evaluation of Lebesgue-Stieltjes integrals is discussed in detail, as are the key theorems of integral calculus as well as the standard convergence theorems. The book then concludes with a brief discussion of multivariate integrals and surveys ok L^p spaces and some applications. Exercises, which extend and illustrate the theory, and provide practice in techniques, are included. Michael Carter and Bruce van Brunt are senior lecturers in mathematics at Massey University, Palmerston North, New Zealand. Michael Carter obtained his Ph.D. at Massey University in 1976. He has research interests in control theory and differential equations, and has many years of experience in teaching analysis. Bruce van Brunt obtained his D.Phil. at the University of Oxford in 1989. His research interests include differential geometry, differential equations, and analysis.
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Table of Contents

1 Real Numbers.- 1.1 Rational and Irrational Numbers.- 1.2 The Extended Real Number System.- 1.3 Bounds.- 2 Some Analytic Preliminaries.- 2.1 Monotone Sequences.- 2.2 Double Series.- 2.3 One-Sided Limits.- 2.4 Monotone Functions.- 2.5 Step Functions.- 2.6 Positive and Negative Parts of a Function.- 2.7 Bounded Variation and Absolute Continuity.- 3 The Riemann Integral.- 3.1 Definition of the Integral.- 3.2 Improper Integrals.- 3.3 A Nonintegrable Function.- 4 The Lebesgue-Stieltjes Integral.- 4.1 The Measure of an Interval.- 4.2 Probability Measures.- 4.3 Simple Sets.- 4.5 Definition of the Integral.- 4.6 The Lebesgue Integral.- 5 Properties of the Integral.- 5.1 Basic Properties.- 5.2 Null Functions and Null Sets.- 5.3 Convergence Theorems.- 5.4 Extensions of the Theory.- 6 Integral Calculus.- 6.1 Evaluation of Integrals.- 6.2 IWo Theorems of Integral Calculus.- 6.3 Integration and Differentiation.- 7 Double and Repeated Integrals.- 7.1 Measure of a Rectangle.- 7.2 Simple Sets and Simple Functions in Two Dimensions.- 7.3 The Lebesgue-Stieltjes Double Integral.- 7.4 Repeated Integrals and Fubini's Theorem.- 8 The Lebesgue SpacesLp.- 8.1 Normed Spaces.- 8.2 Banach Spaces.- 8.3 Completion of Spaces.- 8.4 The SpaceL1.- 8.5 The LebesgueLp.- 8.6 Separable Spaces.- 8.7 ComplexLpSpaces.- 8.8 The Hardy SpacesHp.- 8.9 Sobolev SpacesWk,p.- 9 Hilbert Spaces andL2.- 9.1 Hilbert Spaces.- 9.2 Orthogonal Sets.- 9.3 Classical Fourier Series.- 9.4 The Sturm-Liouville Problem.- 9.5 Other Bases forL2.- 10 Epilogue.- 10.1 Generalizations of the Lebesgue Integral.- 10.2 Riemann Strikes Back.- 10.3 Further Reading.- Appendix: Hints and Answers to Selected Exercises.- References.

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