The narrow Denjoy integral (which also goes by the names Henstock-Kurzweil integral, Perron integral, and Lusin integral) is a transfinite integration process defined by Denjoy in the early 20th century to generalize Lebesgue integration.

In Gordon's 1994 book, "The Integrals of Lebesgue, Denjoy, Perron and Henstock", he defines the Denjoy integral by saying $f:[a,b]\rightarrow\mathbb{R}$ is Denjoy integrable if and only if there is an $F:[a,b]\rightarrow\mathbb{R}$ such that $F$ is generalized absolutely continuous in the restricted sense $(ACG_\ast)$ and $F' = f$ almost everywhere. Then the function $F(x)-F(a)$ may be considered the indefinite Denjoy integral of $f$. Gordon justifies this definition by claiming that Lusin proved the equivalence shortly after Denjoy defined his integral, but gives no source. (Even if he had, such old papers are quite hard to find.)

Question 1: Do you know of an obtainable source that the narrow Denjoy integrals (using Denjoy's original definition) are exactly the $ACG_\ast$ functions?

The wide Denjoy integral (also known as the Khintchine integral), also defined by Denjoy at around the same time, is a slight variation on the narrow Denjoy integral. Its image is the set of generalized absolutely continuous functions $(ACG)$, and as with the narrow version, modern definitions are given in terms of $ACG$ functions (with differentiation replaced by approximate differentiation).

Question 2: Do you know of an obtainable source that the image of the wide Denjoy integral (using the original definition) is the class $ACG$?

I will be extra happy if anyone knows of a single source to reference for both, since the proofs are almost identical.


2 Answers 2


This is not a topic I know much about, but I do have quite a few references pertaining to real analysis and classical point set theory at home, and I looked through these this morning. For many decades the standard reference was Saks' book Theory of the Integral, so I looked there first. I'm fairly certain that everything you want to know can be found in Saks' book, but much of the terminology is a little out-dated and some of the terminology wasn't entirely standard even when the book appeared (or so I've been told). Below are some historical notes from Saks' book that are related to your question. Following these notes are the papers that Saks cites. I've tried to provide fairly full bibliographic information about each paper, as well as provide links leading to freely available digitized copies of the papers on the internet. After the listing of papers cited by Saks, I've included some textbook and other references that I found. I have not had an opportunity to visit a library nor do I have access to Mathematical Reviews or other databases behind paywalls, so it is very likely that my list excludes some "obvious" (to experts in this field) references.

Stanislaw Saks, Theory of the Integral, 2nd revised edition, translated by Laurence Chisholm Young, Monografie Matematyczne #7, G. E. Stechert and Company, 1937, viii + 347 pages. [Reprinted by Dover Publications in 1964 and in 2005.]

The following is from Saks (1937, pp. 214-215):

The first definition of the integral ${\mathcal D}_{*}$ was given in notes dating from 1912 by A. Denjoy [2; 3] who employed the constructive method based on a transfinite process (vide Chap. VIII, § 5). These notes at once attracted the attention of N. Lusin [2] who originated the descriptive theory of this integral. Finally, A. Khintchine [1; 2] and A. Denjoy [4] defined, independently and almost at the same time, the process of integration ${\mathcal D}$ as a generalization of the integral ${\mathcal D}_{*}.$ A systematic account of these researches may be found in the memoir of A. Denjoy [6].

[2] Arnaud Denjoy, Une extension de l'intégrale de M. Lebesgue [An extension of the integral of Mr. Lebesgue], Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences (Paris) 154 (1912), 859-862.

This paper is associated with the session dated 1 April 1912. Reprinted on pp. 341-344 of Denjoy's 1957 Un Demi-Siècle . . . (more details below)

[3] Arnaud Denjoy, Calcul de la primitive de la fonction dérivée la plus générale [Calculation of the primitive of the more general derivative function], Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 154 (1912), 1075-1078.

This paper is associated with the session dated 15 April 1912. Reprinted on pp. 345-347 of Denjoy's 1957 Un Demi-Siècle . . . (more details below)

[4] Arnaud Denjoy, Sur la dérivation et son calcul inverse [On differentiation and its inverse calculation], Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 162 (1916), 377-380. Paper associated with the session dated 13 March 1916.

Reprinted on pp. 348-350 of Denjoy's 1957 Un Demi-Siècle . . . (more details below)

[6] Arnaud Denjoy, Mémoire sur la totalisation des nombres dérivés non sommables [Memoir on the totalization of non-summable derived numbers], Annales Scientifiques de l'École Normale Supérieure (3) 33 / 34 (1916 / 1917), 127-222 / 181-238.

Note: Saks (incorrectly, or intentionally?) lists the pages as 127-222. (NEXT DAY) This morning I happened to look more carefully at my copy of Denjoy's paper and I discovered that its publication is spread over Volumes 33 and 34 of the journal. Moreover, Saks' bibliographic entry for "Denjoy [6]" correctly shows this. I have revised [6] above accordingly. For those who want more details: The part in Volume 33 (1916) consists of pp. 127-222 that were published in consecutive journal issues dated May 1916, June 1916, and July 1916. The part in Volume 34 (1917) consists of pp. 181-238 that were published in consecutive journal issues dated June 1917, July 1917, and August 1917.

Arnaud Denjoy, Un Demi-Siècle (1907-1956) de Notes Communiquées aux Acadèmies ..., Volume II. Le Champ Réel, Gauthier-Villars, 1957.

For Denjoy's 1957 "observations and commentaries" about the 3 papers above, see pp. 73-75 (of a separately paged section) at the end of Volume II.

[1] Aleksandr Yakovlevich Khinchin, Sur une extension de l'intégrale de M. Denjoy [On an extension of the integral of Mr. Denjoy], Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 162 (1916), 287-291.

This paper is associated with the session dated 21 February 1916. A footnote by Hadamard on p. 291 discusses Khinchin's formulation of the generalized Denjoy integral as being independent of Denjoy's formulation of it.

[2] Aleksandr Yakovlevich Khinchin, On the integration process of Denjoy (Russian), Matematicheskii Sbornik 30 #4 (1918), 543-557.

For some reason I saw several references to Khinchin's paper (including the bibliography in Saks' 1937 book) that give the title as Sur le procédé d'intégration de M. Denjoy, with no indication that the paper is actually in Russian and not in French.

[2] Nikolai Nikolaevich Luzin, Sur les propriétés de l'intégrale de M. Denjoy [On the properties of the integral of Mr. Denjoy], Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 155 (1912), 1475-1478.


For historical issues, I recommend Bullen's paper and Pesin's book.

Peter Southcott Bullen, Non-absolute integrals in the twentieth century, AMS Special Session on Nonabsolute Integration (23-24 September 2000), European Mathematical Information Service, The Electronic Library of Mathematics: Mathematical Collections and Conference Proceedings, 27 pages.

V. G. Celidze and A. G. Dzvarseisvili, The Theory of the Denjoy Integral and Some Applications, translated by Peter Southcott Bullen, Series in Real Analysis #3, World Scientific Publishing Company, 1989. xiv + 322 pages.

See title page, copyright page, Contents, Introduction Translator's Preface, Preface to the English Edition

James Michael Foran, Fundamentals of Real Analysis, Monographs and Textbooks in Pure and Applied Mathematics #144, Marcel Dekker, 1991, xiv + 473 pages.

See Chapter 9.1: The Denjoy-Perron Integral (pp. 404-445) and Chapter 9.2: The Denjoy Integral (pp. 445-467).

Ernest William Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier's Series, Volume II, 2nd edition, Cambridge University Press, 1926, x + 780 pages.

Reprinted by Dover Publications in 1958. See Chapter VIII: Non-Absolutely Convergent Integrals, specifically Articles 464-478 (pp. 692-719).

Hyman Kestelman, Modern Theories of Integration, 2nd edition, Dover Publications, 1960, x + 309 pages.

See Chapter IX: Extensions of the Lebesgue Integral, §2. General Denjoy Integral (pp. 217-227).

Henri Léon Lebesgue, Leçons sur l'Intégration et la Recherche des Fonctions Primitives [Lessons on Integration and the Study of Primitive Functions], 2nd edition, Gauthier-Villars, 1928, xv + 342 pages.

See Chapter X: La Totalisation (pp. 202-251).

Isidor Pavlovich Natanson, Theory of Functions of a Real Variable, Volume II, translated by Leo Francis Boron from the 1957 Russian edition, Frederick Ungar Publishing Company, 1960, 265 pages.

See Chapter XVI: Certain Generalizations of the Lebesgue Integral, specifically §7. The Denjoy Integral in the Restricted Sense (i.e. the Denjoy-Perron Integral) (pp. 175-178) and §10. The Denjoy Integral in the Wide Sense (i.e. the Denjoy-Khinchin Integral) (pp. 189-191).

Ivan Nikolaevich Pesin, Classical and Modern Integration Theories, translated and edited by Samuel Kotz, Probability and Mathematical Statistics #8, Academic Press, 1970, xx + 195 pages.

See Chapter 8: The Problem of the Primitive--The Denjoy-Khinchin Integral (pp. 133-159).

  • $\begingroup$ Saks does seem to have this, though it is not set aside as a theorem, at a first look-through I think all the pieces are there. It has a very pleasing parallel analysis of the narrow and wide cases in Chapter VIII, sections 1,4 and 5. Thank you! $\endgroup$ Feb 20, 2014 at 20:01

During my diploma thesis I also found (additionally to references given already by Dave L Renfro) the following quite helpful:

  • Douglas S. Kurtz and Charles W. Swartz, Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and Mcshane, World Scientific Publishing Company, 2004

  • Lee Peng Yee, Henstock Integration, Lanzhou Lectures On Henstock
    Integration, World Scientific Publishing Company, 1989.

  • Lee Peng Yee and Rudolf Vyborny, Integral: An Easy Approach after
    Kurzweil and Henstock (Australian Mathematical Society Lecture
    Series), Cambridge University Press, 2000

Maybe also my thesis is of some interest to you: http://othes.univie.ac.at/16152/ (at least for ACG$_*$ functions).

Edit: I remember only now that Saks' book is available online: http://kielich.amu.edu.pl/Stefan_Banach/e-saks.html

  • 1
    $\begingroup$ Thanks for the link to the 1937 edition of Saks book. I knew the 1933 1st edition (in French) was freely available on-line, but I didn't know that the 1937 edition was freely available. I've included your link into my answer. $\endgroup$ Feb 19, 2014 at 22:14

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