The Lebesgue-Stieltjes Integral: A Practical Introduction

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.