目录
 前辅文
 1 An Historical Overview
  1.1 Rearrangements
  1.2 The Lune of Hippocrates
  1.3 Eudoxus and the Method of Exhaustion
  1.4 Archimedes’ Method
  1.5 Gottfried Leibniz and Isaac Newton
  1.6 Augustin-Louis Cauchy
  1.7 Bernhard Riemann
  1.8 Thomas Stieltjes
  1.9 Henri Lebesgue
  1.10 The Lebesgue-Stieltjes Integral
  1.11 Ralph Henstock and Jaroslav Kurzweil
  1.12 Norbert Wiener
  1.13 Richard Feynman
  1.14 References
 2 The Cauchy Integral
  2.1 Exploring Integration
  2.2 Cauchy’s Integral
  2.3 Recovering Functions by Integration
  2.4 Recovering Functions by Differentiation
  2.5 A Convergence Theorem
  2.6 Joseph Fourier
  2.7 P. G. Lejeune Dirichlet
  2.8 Patrick Billingsley’s Example
  2.9 Summary
  2.10 References
 3 The Riemann Integral
  3.1 Riemann’s Integral
  3.2 Criteria for Riemann Integrability
  3.3 Cauchy and Darboux Criteria for Riemann Integrability
  3.4 Weakening Continuity
  3.5 Monotonic Functions Are Riemann Integrable
  3.6 Lebesgue’s Criteria
  3.7 Evaluating à la Riemann
  3.8 Sequences of Riemann Integrable Functions
  3.9 The Cantor Set (1883)
  3.10 A Nowhere Dense Set of Positive Measure
  3.11 Cantor Functions
  3.12 Volterra’s Example
  3.13 Lengths of Graphs and the Cantor Function
  3.14 Summary
  3.15 References
 4 The Riemann-Stieltjes Integral
  4.1 Generalizing the Riemann Integral
  4.2 Discontinuities
  4.3 Existence of Riemann-Stieltjes Integrals
  4.4 Monotonicity of ø
  4.5 Euler’s Summation Formula
  4.6 Uniform Convergence and R-S Integration
  4.7 References
 5 Lebesgue Measure
  5.1 Lebesgue’s Idea
  5.2 Measurable Sets
  5.3 Lebesgue Measurable Sets and Caratheodory
  5.4 Sigma Algebras
  5.5 Borel Sets
  5.6 Approximating Measurable Sets
  5.7 Measurable Functions
  5.8 More Measureable Functions
  5.9 What Does Monotonicity Tell Us?
  5.10 Lebesgue’s Differentiation Theorem
  5.11 References
 6 The Lebesgue Integral
  6.1 Introduction
  6.2 Integrability: Riemann Ensures Lebesgue
  6.3 Convergence Theorems
  6.4 Fundamental Theorems for the Lebesgue Integral
  6.5 Spaces
  6.6 L2[-π, π] and Fourier Series
  6.7 Lebesgue Measure in the Plane and Fubini’s Theorem
  6.8 Summary
  6.9 References
 7 The Lebesgue-Stieltjes Integral
  7.1 L-S Measures and Monotone Increasing Functions
  7.2 Carathéodory’s Measurability Criterion
  7.3 Avoiding Complacency
  7.4 L-S Measures and Nonnegative Lebesgue Integrable Functions
  7.5 L-S Measures and Random Variables
  7.6 The Lebesgue-Stieltjes Integral
  7.7 A Fundamental Theorem for L-S Integrals
  7.8 Reference
 8 The Henstock-Kurzweil Integral
  8.1 The Generalized Riemann Integral
  8.2 Gauges and δ-fine Partitions
  8.3 H-K Integrable Functions
  8.4 The Cauchy Criterion for H-K Integrability
  8.5 Henstock’s Lemma
  8.6 Convergence Theorems for the H-K Integral
  8.7 Some Properties of the H-K Integral
  8.8 The Second Fundamental Theorem
  8.9 Summary
  8.10 References
 9 The Wiener Integral
  9.1 Brownian Motion
  9.2 Construction of the Wiener Measure
  9.3 Wiener’s Theorem
  9.4 Measurable Functionals
  9.5 The Wiener Integral
  9.6 Functionals Dependent on a Finite Number of t Values
  9.7 Kac’s Theorem
  9.8 References
 10 The Feynman Integral
  10.1 Introduction
  10.2 Summing Probability Amplitudes
  10.3 A Simple Example
  10.4 The Fourier Transform
  10.5 The Convolution Product
  10.6 The Schwartz Space
  10.7 Solving Schrödinger Problem A
  10.8 An Abstract Cauchy Problem
  10.9 Solving in the Schwartz Space
  10.10 Solving Schrödinger Problem B
  10.11 References
 Index
 About the Author
 前辅文
 1 An Historical Overview
  1.1 Rearrangements
  1.2 The Lune of Hippocrates
  1.3 Eudoxus and the Method of Exhaustion
  1.4 Archimedes’ Method
  1.5 Gottfried Leibniz and Isaac Newton
  1.6 Augustin-Louis Cauchy
  1.7 Bernhard Riemann
  1.8 Thomas Stieltjes
  1.9 Henri Lebesgue
  1.10 The Lebesgue-Stieltjes Integral
  1.11 Ralph Henstock and Jaroslav Kurzweil
  1.12 Norbert Wiener
  1.13 Richard Feynman
  1.14 References
 2 The Cauchy Integral
  2.1 Exploring Integration
  2.2 Cauchy’s Integral
  2.3 Recovering Functions by Integration
  2.4 Recovering Functions by Differentiation
  2.5 A Convergence Theorem
  2.6 Joseph Fourier
  2.7 P. G. Lejeune Dirichlet
  2.8 Patrick Billingsley’s Example
  2.9 Summary
  2.10 References
 3 The Riemann Integral
  3.1 Riemann’s Integral
  3.2 Criteria for Riemann Integrability
  3.3 Cauchy and Darboux Criteria for Riemann Integrability
  3.4 Weakening Continuity
  3.5 Monotonic Functions Are Riemann Integrable
  3.6 Lebesgue’s Criteria
  3.7 Evaluating à la Riemann
  3.8 Sequences of Riemann Integrable Functions
  3.9 The Cantor Set (1883)
  3.10 A Nowhere Dense Set of Positive Measure
  3.11 Cantor Functions
  3.12 Volterra’s Example
  3.13 Lengths of Graphs and the Cantor Function
  3.14 Summary
  3.15 References
 4 The Riemann-Stieltjes Integral
  4.1 Generalizing the Riemann Integral
  4.2 Discontinuities
  4.3 Existence of Riemann-Stieltjes Integrals
  4.4 Monotonicity of ø
  4.5 Euler’s Summation Formula
  4.6 Uniform Convergence and R-S Integration
  4.7 References
 5 Lebesgue Measure
  5.1 Lebesgue’s Idea
  5.2 Measurable Sets
  5.3 Lebesgue Measurable Sets and Caratheodory
  5.4 Sigma Algebras
  5.5 Borel Sets
  5.6 Approximating Measurable Sets
  5.7 Measurable Functions
  5.8 More Measureable Functions
  5.9 What Does Monotonicity Tell Us?
  5.10 Lebesgue’s Differentiation Theorem
  5.11 References
 6 The Lebesgue Integral
  6.1 Introduction
  6.2 Integrability: Riemann Ensures Lebesgue
  6.3 Convergence Theorems
  6.4 Fundamental Theorems for the Lebesgue Integral
  6.5 Spaces
  6.6 L2[-π, π] and Fourier Series
  6.7 Lebesgue Measure in the Plane and Fubini’s Theorem
  6.8 Summary
  6.9 References
 7 The Lebesgue-Stieltjes Integral
  7.1 L-S Measures and Monotone Increasing Functions
  7.2 Carathéodory’s Measurability Criterion
  7.3 Avoiding Complacency
  7.4 L-S Measures and Nonnegative Lebesgue Integrable Functions
  7.5 L-S Measures and Random Variables
  7.6 The Lebesgue-Stieltjes Integral
  7.7 A Fundamental Theorem for L-S Integrals
  7.8 Reference
 8 The Henstock-Kurzweil Integral
  8.1 The Generalized Riemann Integral
  8.2 Gauges and δ-fine Partitions
  8.3 H-K Integrable Functions
  8.4 The Cauchy Criterion for H-K Integrability
  8.5 Henstock’s Lemma
  8.6 Convergence Theorems for the H-K Integral
  8.7 Some Properties of the H-K Integral
  8.8 The Second Fundamental Theorem
  8.9 Summary
  8.10 References
 9 The Wiener Integral
  9.1 Brownian Motion
  9.2 Construction of the Wiener Measure
  9.3 Wiener’s Theorem
  9.4 Measurable Functionals
  9.5 The Wiener Integral
  9.6 Functionals Dependent on a Finite Number of t Values
  9.7 Kac’s Theorem
  9.8 References
 10 The Feynman Integral
  10.1 Introduction
  10.2 Summing Probability Amplitudes
  10.3 A Simple Example
  10.4 The Fourier Transform
  10.5 The Convolution Product
  10.6 The Schwartz Space
  10.7 Solving Schrödinger Problem A
  10.8 An Abstract Cauchy Problem
  10.9 Solving in the Schwartz Space
  10.10 Solving Schrödinger Problem B
  10.11 References
 Index
 About the Author