The Riemann Approach to Integration
Local Geometric Theory

A detailed exposition of generalised Riemann–Stieltjes integrals.

Washek F. Pfeffer (Author)

9780521440356, Cambridge University Press

Hardback, published 25 February 1994

324 pages
23.7 x 16 x 2.7 cm, 0.629 kg

"The author presents a detailed exposition of some recent (Riemann!) approaches to generalized integrals of real-valued functions on compact intervals in Euclidean spaces Rm (m>=1)...The author presents this theory in a very complete, thorough manner." R.G. Bartle, Mathematical Reviews

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.

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.

Subject Areas: Calculus & mathematical analysis [PBK]