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The Riemann Approach to Integration

The Riemann Approach to Integration
Local Geometric Theory

Part of Cambridge Tracts in Mathematics

  • Date Published: March 1994
  • availability: Available
  • format: Hardback
  • isbn: 9780521440356

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  • This book presents a detailed and mostly elementary exposition of the generalised Riemann-Stieltjes integrals discovered by Henstock, Kurzweil, and McShane. Along with the classical results, it contains some recent developments connected with lipeomorphic change of variables and the divergence theorem for discontinuously differentiable vector fields. Defining the Lebesgue integral in Euclidean spaces from the McShane point of view has a clear pedagogical advantage: the initial stages of development are both conceptually and technically simpler. The McShane integral evolves naturally from the initial ideas about integration taught in basic calculus courses. The difficult transition from subdividing the domain to subdividing the range, intrinsic to the Lebeque definition, is completely bypassed. The unintuitive Caratheodory concept of measurability is also made more palatable by means of locally fine partitions. Although written as a monograph, the book can be used as a graduate text, and certain portions of it can be presented even to advanced undergraduate students with a working knowledge of limits, continuity and differentiation on the real line.

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    Product details

    • Date Published: March 1994
    • format: Hardback
    • isbn: 9780521440356
    • length: 324 pages
    • dimensions: 237 x 160 x 27 mm
    • weight: 0.629kg
    • availability: Available
  • Table of Contents

    Preface
    Acknowledgments
    Part I. One-Dimensional Integration:
    1. Preliminaries
    2. The McShane integral
    3. Measure and measurability
    4. Integrable functions
    5. Descriptive definition
    6. The Henstock-Kurzweil integral
    Part II. Multi-Dimensional Integration:
    7. Preliminaries
    8. The McShane integral
    9. Descriptive definition
    10. Change of variables
    11. The gage integral
    12. The F-integral
    13. Recent developments
    Bibliography
    List of symbols
    Index.

  • Author

    Washek F. Pfeffer, University of California, Davis

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