The Lebesgue-Stieltjes Integral: A Practical Introduction

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