The Lebesgue-Stieltjes Integral

1. Front Matter

   Pages i-ix

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2. Real Numbers

   • M. Carter, B. van Brunt

   Pages 1-10

3. Some Analytic Preliminaries

   • M. Carter, B. van Brunt

   Pages 11-38

4. The Riemann Integral

   • M. Carter, B. van Brunt

   Pages 39-48

5. The Lebesgue-Stieltjes Integral

   • M. Carter, B. van Brunt

   Pages 49-70

6. Properties of the Integral

   • M. Carter, B. van Brunt

   Pages 71-85

7. Integral Calculus

   • M. Carter, B. van Brunt

   Pages 87-111

8. Double and Repeated Integrals

   • M. Carter, B. van Brunt

   Pages 113-122

9. The Lebesgue Spaces L^p

   • M. Carter, B. van Brunt

   Pages 123-163

10. Hilbert Spaces and L²

    • M. Carter, B. van Brunt

    Pages 165-202

11. Epilogue

    • M. Carter, B. van Brunt

    Pages 203-208

12. Back Matter

    Pages 209-230

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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. His publications include

• Fourier series
• Lebesgue Theory
• Lebesgue-Stieltjes Integral
• Riemann integral
• calculus
• differential equation
• differential geometry
• measure

• Institute of Fundamental Sciences Palmerston North Campus, Massey University, New Zealand

  M. Carter, B. Brunt

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