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The integrals of Lebesgue, Denjoy, Perron, and Henstock. (English) Zbl 0807.26004
Graduate Studies in Mathematics. 4. Providence, RI: American Mathematical Society (AMS). xi, 395 p. (1994).
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 S. Lu and K. Liao [Real Anal. Exch. 16, No. 1, 74-78 (1991; Zbl 0744.26009)] and K. Liao and T.-S. Chew [Real Anal. Exch. 19, No. 1, 81-97 (1994; Zbl 0802.26005)], respectively.

26A39 Denjoy and Perron integrals, other special integrals
26-02 Research exposition (monographs, survey articles) pertaining to real functions
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration