The Riemann approach to integration : local geometric theory
著者
書誌事項
The Riemann approach to integration : local geometric theory
（Cambridge tracts in mathematics, 109）
Cambridge University Press, 2008
- : pbk
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注記
"First publushed 1993, this digitally printed version 2008"--t.p.verso
Includes bibliographical references (p. [293]-295) and index
内容説明・目次
内容説明
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.
目次
- 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.
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