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What did Riemann do for the theory of integration?

I am asking because I hear his name a lot in relation to integration and it is often implied that he made large contributions, but I do not know what they are.

To clarify, Cauchy was the first to make the integral rigourous. Then, Riemann then gave a slightly different definition of the integral. Then, Darboux gave a defintion that simplified the proofs to the common theorems of integration. It should also be noted that what is often taught as the Riemann integral is normally the Darboux integral.

Did Riemann do more for integration than this slight different defintion of the integral, whose main ideas were preceeded by Cauchy and whose related proofs were simplified by Darboux?

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The following is a slightly edited version of my 31 January 2003 sci.math post archived at Math Forum.

Riemann [6] introduced his integral in his December 1853 Habilitationsschrift thesis. In his thesis he also gave an example, correctly verified, of a Riemann integrable function whose discontinuities form a dense set. Riemann's thesis wasn't widely known until it was published (by Dedekind in 1867) shortly after his death in 1866. Two years later in 1870, Hankel [3] proved that every Riemann integrable function has a dense set of continuity points. (For more about Hankel's result, see Renfro [5].)

Thus, within TWO years after the publication of Riemann's definition of the integral, it was known that Riemann integrable functions are very well behaved in one sense and it was known that they can be very badly behaved in another sense. They are well behaved in the sense that they have a dense set of continuity points, and thus one might think they are so close to being continuous that they can't be any worse behaved than functions such as piecewise continuous functions, the characteristic function of $\{\frac{1}{2}, \; \frac{1}{3}, \; \frac{1}{4}, \; \ldots \},$ etc. On the other hand, they can be so badly behaved that they can have a dense set of discontinuity points, and thus it would not be intuitively evident to a novice that such a function could be continuous anywhere, except maybe at isolated points or on sets such as $\{\frac{1}{2}, \; \frac{1}{3}, \; \frac{1}{4}, \; \ldots \}.$

Incidentally, the definition that Riemann gave for his integral is a minor modification of Cauchy's definition from 1823. Cauchy used partitions whose associated subintervals have varying lengths, but Cauchy used only left (or only right) endpoints of these subintervals at which to evaluate the function. In fact, Cauchy's seemingly more restrictive definition actually gives rise to the same collection of integrable functions that Riemann's definition does --- see Gillespie [2].

Riemann's nontrivial contributions to this topic were: (A) giving a necessary and sufficient condition for integrability based on the behavior of a function; (B) using this condition to prove the integrability of a certain function having a dense set of discontinuities; (C) putting the focus on the collection of functions that are integrable according to some notion of integrability, rather than defining a notion of integrability only for the purpose of being able to prove certain desired integrability properties. Regarding (B), I believe this was the first time a function that was continuous on a dense set and discontinuous on another dense set had been defined (or even contemplated, for that matter). A well known example of such a function is the ruler function, which is also called the Thomae function because it first appeared in an 1875 booklet by Thomae.

Cauchy, I believe, only proved that functions having at most finitely many discontinuities in every interval are integrable. Moreover, these were the only functions that Cauchy was interested in, and so Cauchy never sought to generalize his results to more pathological functions nor did he consider his definition of the integral as defining a collection of functions that might be worthy of study.

The measure zero version for Riemann integrability was obtained independently by Lebesgue [4] (p. 24; p. 254 of published version) and Vitali [8] [9] and Young [11].

The Lebesgue/Vitali/Young integrability condition is actually a small step from results already known by 1875. The integrability condition that Riemann gave, what I called contribution (A) above, involved the oscillation of a function in an interval. In 1870 Hankel [3] reformulated Riemann's condition in terms of the oscillation of a function at a point, a notion that was also first introduced in this paper. Hankel's reformulated condition is a bit closer to the notion of continuity at a point, since (as Hankel was well aware) it is straightforward to show that $f$ is continuous at $x=a$ if and only if the oscillation of $f$ at $x=a$ is zero.

Specifically, Hankel said that a function $f$ is Riemann integrable if and only if for each $\epsilon > 0$ the set of points at which $f$ has an oscillation greater than $\epsilon$ can be covered by a finite collection of intervals whose lengths have an arbitrarily small sum. [This differs from the way Lebesgue measure zero sets are defined only in that finitely many intervals, rather than countably many intervals, are allowed for the coverings.] Hankel's claim is true. However, Hankel's proof that a Riemann integrable function has this property was incorrect. [The other half of Hankle's "iff" claim, namely that this property implies Riemann integrability, was correctly proved by Hankel.] Hankel's error was that he tried to prove this by trying to prove a stronger result --- he tried to prove that every function whose points of continuity form a dense set will be Riemann integrable. Unfortunately, this last result is too strong, and not just because the methods of proof Hankel used were too weak, but in fact the result itself is not even true! The characteristic function of a Cantor set with positive measure is a counterexample. It was precisely to show this error in Hankel's paper that Smith [7] used a Cantor set of positive measure in his 1875 paper, the paper that contained the first ever construction of a Cantor set (whether of measure zero or of positive measure, and Smith gave both kinds).

Hankel's proof was repaired by Ascoli [1] in 1875. An even more precise version was given by Volterra [10] in 1881. We say that a set $E$ is "Jordan null" (or has Jordan content zero) if, given any $\delta > 0,$ there exists a finite collection of intervals covering $E$ whose lengths have a sum less than $\delta.$ Then, using this notion, which I might add is equivalent (when $E$ is bounded) to the closure of $E$ having Lebesgue measure zero), Hankel's integrability condition says that $f$ is Riemann integrable if and only if for each $\epsilon > 0$ the points at which $f$ has an oscillation greater than $\epsilon$ form a Jordan null set. Volterra showed that the result continues to hold if we use the "left oscillation of $f$" rather than the oscillation of $f,$ or equivalently (consider $-f),$ the right oscillation of $f.$ [Left and right oscillation at a point are defined in the obvious way.]

The jump from these results to Lebesgue's result is small once the notion of a set of measure zero had been considered worthy of consideration, since it's essentially a matter of a countable union of measure zero sets having measure zero. To be explicit, we can obtain a covering of the discontinuities of $f$ with intervals whose lengths have a sum less than $\delta$ by taking the union over $n = 1,\; 2, \;3, \; \ldots$ of the following finite collections $\mathcal{C}_n$ of intervals that we know exist by Hankel's condition -- Given $n,$ let $\mathcal{C}_n$ be a finite collection of intervals covering the points where $f$ has an oscillation greater than $\frac{1}{n}$ and such that the sum of the lengths of the intervals in $\mathcal{C}_n$ is less than $\frac{\delta}{2^n}.$

[1] Giulio Ascoli, Sul concetto di integrale definito [On the notion of definite integral], Atti della Accademia Reale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali (2) 2 (1875), 862-872.

[2] David Clinton Gillespie, The Cauchy definition of a definite integral, Annals of Mathematics (2) 17 #2 (December 1915), 61-63.

[3] Hermann Hankel, Untersuchungen über die unendlich oft oszillierenden und unstetigen funktionen [Investigations of functions with infinitely many oscillations and discontinuous functions], Memoir presented at the University of Tübingen on 6 March 1870, 51 pages.

Reprinted in Mathematische Annalen 20 #1 (1882), 63-112 (google books version; Göttinger Digitalisierungszentrum version). A French translation by Jeanne Peiffer, with some additional remarks by Pierre Dugac and René Taton, appears in Cahiers du Séminaire d'Histoire des Mathématiques 9 (1988), 139-209.

[4] Henri Léon Lebesgue, Intégrale, Longueur, Aire [Integral, Length, Area], Ph.D. dissertation (under Félix Edouard Justin Émile Borel), Université Henri Poincaré Nancy 1, 1902, iv + 129 pages.

The published version (Springer pay wall version; another copy) is in Annali di Mathematica Pura ed Applicata (3) 7 (1902), 231-359.

[5] Dave L. Renfro, 28 January 2003 Historia-Matematica post archived at Math Forum.

[6] Georg Friedrich Bernhard Riemann, Ueber die Darstellbarkeit einer Function durch eine Trigonometrische Reihe [On the Representability of a Function by a Trigonometric Series], Habilitationsschrift thesis (under Johann Carl Friedrich Gauss), Göttingen University, December 1853.

The original version of Riemann’s thesis is a handwritten manuscript that is currently in Riemann’s Nachlass at the Göttingen University Library Archives. Before 1868 it is likely that only a few people had seen Riemann’s thesis (mainly Berlin mathematicians, and possibly some others while visiting Berlin), with perhaps a few others having heard about its contents by word of mouth. The first printed and circulated version was published in Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1867), 87-131. This first publication was due to the efforts of Julius Wilhelm Richard Dedekind. Although Dedekind’s introductory footnote on the first page is dated July 1867 and the date given on the title page is 1867, I believe that printed copies did not actually reach the mathematical public until 1868. A French translation (by Jean Gaston Darboux and Guillaume Jules Hoüel), Sur la possibilité de représenter une fonction par une série trigonométrique, was published in Bulletin des Sciences Mathématiques et Astronomiques (1) (1873), 20-48 & 79-96. English translations can be found in Bernhard Riemann. Collected Papers (2004, pp. 219-256) and God Created the Integers (2005, pp. 826-865; 2007, pp. 992-1031).

[7] Henry John Stephen Smith, On the integration of discontinuous functions, Proceedings of the London Mathematical Society (1) 6 (1875), 140-153.

[8] Giuseppe Vitali, Sulla condizione di integrabilità delle funzioni [On the integrability condition of functions], Bullettino Mensile della Accademia Gioenia di Scienze Naturali in Catania (N.S.) 79 (1903), 27-30.

[9] Giuseppe Vitali, Sulla integrabilità delle funzioni [On the integrability of functions], Rendiconti del Reale Istituto Lombardo di Scienze e Lettere (2) 37 (1904), 69-73.

[10] Vito Volterra, Alcune osservazioni sulle funzioni punteggiate discontinue [Some observations on pointwise discontinuous functions], Giornale di Matematiche di Battaglini 19 (1881), 76-86.

[11] William Henry Young, A note on the condition of integrability of a function of a real variable, Quarterly Journal of Pure and Applied Mathematics 35 (1904), 189-192.

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Cauchy made integral rigorous, and proved that integral (in the sense of Cauchy) exists for continuous functions. Riemann proposed a more general definition, (integral in the sense of Riemann) and introduced the new class of functions, which are called now Riemann-integrable. This class is strictly larger than continuous (or piecewise-continuous) functions. Later, after Riemann, necessary and sufficient conditions were proved for a function to be Riemann-integrable. (A bounded function is Riemann-integrable if and only if it is continuous almost everywhere).

Sourcs: I. N. Pesin, Classical and modern integration theories, Acad. Press, 1970. (Translated from the Russian original, 1966), Wikipedia.

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