Author | Carter, M. author |
---|---|
Title | The Lebesgue-Stieltjes Integral [electronic resource] : A Practical Introduction / by M. Carter, B. van Brunt |
Imprint | New York, NY : Springer New York : Imprint: Springer, 2000 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-1174-7 |
Descript | IX, 230 p. online resource |
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 Fubiniโs 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