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Topics in Banach space integration. (English) Zbl 1088.28008
Series in Real Analysis 10. Hackensack, NJ: World Scientific (ISBN 981-256-428-4/hbk). 312 p. (2005).
In the late 1950’s and early 1960’s J. Kurzweil and R. Henstock introduced an elementary generalization of the Riemann integral based on Riemann sums which generalized the Lebesgue integral and was actually equivalent to the Perron/Denjoy integral. Later, E. J. McShane gave a slight alteration of the Henstock/Kurzweil integral which was shown to be equivalent to the Lebesgue integral. It is straightforward to generalize both the Henstock/Kurzweil and McShane integrals to functions which have values in a Banach space. The authors present these generalizations and then develop their basic properties including several interesting convergence theorems for the McShane integral. The authors also give developments of the Bochner, Dunford and Pettis integrals for Banach space valued functions. They then proceed to compare the relationships between the various integrals. For example, they show that a Banach space valued function is McShane integrable iff the function is both Pettis and Henstock/Kurzweil integrable. The authors also define strong versions of both the Henstock/Kurzweil and McShane integrals and show that the strong McShane integral is equivalent to the Bochner integral. For functions defined on a bounded interval in the line, the authors give characterizations of both the strong Henstock/Kurzweil and strong McShane integrals in terms of their primitives or indefinite integrals. The last chapter in the book discusses extensions of the Bochner, Dunford and Pettis integrals.
This book is carefully written and should be accessible to anyone with a basic knowledge of classical integration theory and elementary functional analysis. The book contains an extensive bibliography and should be useful to those with interests in Banach space integration.

##### MSC:
 28B05 Vector-valued set functions, measures and integrals 28-02 Research exposition (monographs, survey articles) pertaining to measure and integration 46G10 Vector-valued measures and integration 46G12 Measures and integration on abstract linear spaces 26A39 Denjoy and Perron integrals, other special integrals