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  • 1. Citations From References: 0 Article From Reviews: 0 MR2388593 26A42 (39A10) Thomson, Brian S. (3-SFR) Henstock-Kurzweil integrals on time scales. (English summary) Panamer. Math. J. 18 (2008), no. 1, 119. {There will be no review of this item.} c Copyright American Mathematical Society 2009 Citations From References: 0 Article From Reviews: 0 MR2384585 (2009a:26006) 26A39 (01A60 26A42) Thomson, Brian S. (3-SFR) The natural integral on the real line. Sci. Math. Jpn. 67 (2008), no. 1, 2335. This essay is dedicated to Ralph Henstock (19232007) and his theory of integration. The author argues that a Riemann-type integral on the real line which was introduced by Henstock (and at the same time by Czech mathematician J. Kurzweil) and which covers the Lebesgue integral, is not just easier to present than the last oneit is easier to present than the Riemann integral. The name for this integral proposed in the title of the paper is considered by the author as a suitable brand in the mathematical market which could persuade the academic community to introduce this theory into the undergraduate curriculum. The notions of the covering relations and the one of the integration basis introduced by the author of the paper are used to give an account of Henstocks ideas on the way to generalize his denition in a more abstract setting. A program for developing the integration theory is sketched. Reviewed by V. A. Skvortsov c Copyright American Mathematical Society 2009 Citations From References: 0 Article From Reviews: 0
  • 2. MR2321251 (2008c:26001) 26-01 (26A06 97D80) Thomson, Brian S. (3-SFR) Rethinking the elementary real analysis course. Amer. Math. Monthly 114 (2007), no. 6, 469490. This is an important paper for mathematical education and should be read by anyone planning a course or about to write a text on analysis or calculus. Ever since Henstock rather brashly said Lebesgue is dead! at the Stockholm meeting in 1962 there have been many hardy souls, colleagues, students and disciples of either Henstock or Kurzweil suggesting the more reasonable and practical Riemann is dead!. Many papers have been written that make this point, as well as books by both Henstock and Kurzweil, the author of the present paper, Bartle, DePree and Schwarz, Leader, Lee and V born , Mawhin, McLeod, McShane, and no doubt others; as well y y Dieudonn , in the book that is quoted at the beginning of the present article, made the same point e from a completely different point of view. All argue that by now the teaching of both the Riemann and Riemann-Stieltjes integrals should cease; these integrals should be replaced by the Newton and Cauchy integrals in elementary calculus courses and by the generalized Riemann integral in elementary analysis courses, the latter being taught in a way that would lead into Lebesgue theory in more advanced courses. To date the conservative nature of academia has not heard these arguments. The present paper suggests a very attractive method for the elementary analysis course mentioned above. The author has a rather extreme aversion to the plethora of gauges and tags that are the norm in most approaches to the generalized Riemann integral. Instead he suggests that the basis of the analysis course should be Cousins Lemma. This very elementary formulation of the completeness axiom of the real line allows for proofs of all the properties normally deduced from that axiom, including the properties of continuous functions normally considered in basic analysis courses proofs that are both simple and transparent. It further leads very naturally into the generalized Riemann integral and later to measure theory if that is desired. The paper is very clearly written and is very persuasive but is not an easy read, especially towards the end, and the reader may need the help of a standard book on the generalized Riemann integral. In addition the author shows that a simple recent extension of Cousins Lemma will allow some very neat proofs of more subtle properties that may or may not be appropriate for the rst analysis course but certainly would allow for the progression to a more advanced course and to measure theory. It is difcult to change a very well-established academic and publishing tradition but the reviewer hopes that this article will start at least the beginnings of a change. There are already non- or semi- commercial web texts that are moving in this direction, some under the inuence of the Dump-the- Riemann-Integral-Project (see and others less radical being produced by the Trillia Group ( These should all be explored by anyone teaching in this eld. Reviewed by P. S. Bullen References 1. R. G. Bartle, A Modern Theory of Integration, Graduate Studies in Math., no. 32, American
  • 3. Mathematical Society, Providence, 2001. MR1817647 (2002d:26001) 2. , Return to the Riemann integral, this Monthly 103 (1996) 625632. MR1413583 (97h:26007) 3. J. Hagood, The Lebesgue differentiation theorem via nonoverlapping interval covers, Real Anal. Exch. 29 (200304) 953956. MR2083830 (2005d:26007) 4. J. Hagood and B. S. Thomson, Recovering a function from a Dini derivative, this Monthly [to appear]. cf. MR 2006i:26010 5. R. Henstock, The efciency of convergence factors for functions of a continuous real variable, J. London Math. Soc. 30 (1955) 273286. MR0072968 (17,359f) 6. R. Henstock, A Riemann-type integral of Lebesgue power, Canad. J. Math. 20 (1968) 7987. MR0219675 (36 #2754) 7. J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a pa- rameter, Czechoslovak Math. J. 7 (1957) 418449. MR0111875 (22 #2735) 8. S. Leader, What is a differential? A new answer from the generalized Riemann integral, this Monthly 93 (1986) 348356. MR0841112 (87e:26002) 9. S. Leader, The Kurzweil-Henstock Integral and Its Differentials. A Unied Theory of Integration on R and Rn , Marcel Dekker, New York, 2001. MR1837270 (2002i:26005) 10. H. Lebesgue, Lecons sur LInt gration, Chelsea, New York, 1973; reprint of the 1928 Paris e 2nd ed. 11. J. Mawhin, Introduction a lanalyse, 2nd. Cabay Libraire-Editeur S.A., Louvain-la-Neuve, France, 1981. MR0631523 (83h:26003) 12. E. J. McShane, A unied theory of integration, this Monthly 80 (1973) 349359. MR0318434 (47 #6981) 13. E. J. McShane, Unied Integration, Academic, New York, 1983. MR0740710 (86c:28002) 14. A. Smithee, The Integral Calculus, available at http://www.classicalrealanalysis. com. 15. B. S. Thomson, On full covering properties, Real Anal. Exchange 6 (1980/81) 7793. MR0606543 (82c:26008) 16. E. Zakon, Mathematical Analysis, available at Note: This list reects references listed in the original paper as accurately as possible with no attempt to correct errors. c Copyright American Mathematical Society 2008, 2009 Citations From References: 1 Article From Reviews: 1
  • 4. MR2202919 (2006i:26010) 26A42 (26A24 26A39) Hagood, John W. (1-NAZ-MS); Thomson, Brian S. (3-SFR) Recovering a function from a Dini derivative. Amer. Math. Monthly 113 (2006), no. 1, 3446. The inversion formula b F (b) F (a) = D+ F (x) dx a for a function F having nite upper right-hand Dini derivative D+ F (x) at each x R is discussed. The authors seek a suitable Riemann-type denition of the integral to obtain this formula without an integrability assumption for D+ F . To this purpose, the notion of so-called right full cover (a special case of covering relation due to the second author [B. S. Thomson, Mem. Amer. Math. Soc. 93 (1991), no. 452, vi+96 pp.; MR1078198 (92d:26002)]) is introduced. The inversion formula is established for any continuous F , with the integral being understood as the lower (Henstock- Kurzweil-type) integral dened with respect to right full covers. Clearly, an analogous result holds