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Zbl 0807.26004
Gordon, Russell A.
The integrals of Lebesgue, Denjoy, Perron, and Henstock. (English)
[B] Graduate Studies in Mathematics. 4. Providence, RI: American Mathematical Society (AMS). xi, 395 p. \$ 59.00 (1994). ISBN 0-8218-3805-9/hbk

The Denjoy-Perron-Henstock integral is an extension of the Lebesgue integral which integrates the derivatives. The author gives an elementary account of the theory via Lebesgue measure. The approach is entirely classical and confined to functions defined on a compact interval on the real line. The author begins with the Lebesgue theory, develops each of the three extensions separately, and proves their equivalence and various properties including integration by parts and convergence theorems. The general Denjoy (Khintchine) integral and the approximately continuous Perron (AP) integral of Burkill are also included. Furthermore, the book has a chapter on Darboux functions and it contains a result of Tolstov on the AP integral which is not easily available elsewhere. The two problems left open in the book on convergence theorems and the approximately continuous Denjoy integral have been completely answered in {\it S. Lu} and {\it K. Liao} [Real Anal. Exch. 16, No. 1, 74-78 (1991; Zbl 0744.26009)] and {\it K. Liao} and {\it T.-S. Chew} [Real Anal. Exch. 19, No. 1, 81-97 (1994; Zbl 0802.26005)], respectively.
[Lee Peng-Yee (Singapore)]
MSC 2000:
*26A39 Special integrals of functions of one real variable
26-02 Research monographs (real functions)
28-02 Research monographs (measure and integration)

Keywords: Khintchine integral; approximately continuous Perron integral; Denjoy- Perron-Henstock integral; integration by parts; convergence theorem; Darboux functions; AP integral; approximately continuous Denjoy integral

Citations: Zbl 0744.26009; Zbl 0802.26005

Cited in: Zbl 1225.26016

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