Modern Integration

Adam Stasiak
History of Mathematics, Spring 2000


Bernhard Riemann and Henri Lebesgue are responsible for the most recent and widely used refinements of integration theory. They both continued to extend the work of more than a millennia of mathematicians, providing successively more general integration theories, allowing the study of a wider range of functions and a greater variety of applications.


The earliest notable use of a form of integration is due to the Greeks, who sought the areas of difficult geometric shapes, such as circles and crescents or "lunes". The Greek mathematician Hippocrates, about 430 B.C., was one of the earliest to find the area of a planar figure bounded by arcs rather than straight lines. However his method was chiefly geometric and relied on the observation that the areas of sectors of a circle vary with the square of the radii. As an example, we consider the arc ABC consisting of half of a circle. If we place the arc AFC so that F lies within the circle, and its center is on the circle at a point D opposite B, then the area between ABC and AFC will be equal to that of a square with side of the same length as the radius of the circle.

Significantly more interesting is the work of Eudoxus and its application by Archimedes. Eudoxus conceived an axiom that, at its heart, is analogous to much later work by Cauchy in the 18th century. The axiom of Eudoxus is: "Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continuously, there will be left some magnitude which will be less than the lesser magnitude set out." (Burk 4) Examination of this shows it is equivalent to selecting some epsilon and then removing pieces of the area of at least half that which remains. Then the limit of this process will be the area of the figure. Archimedes used this, along with some clever geometric reasoning to discover that the area of a parabolic segment is two- thirds that of its circumscribed triangle. (Burk 6)

The next work of note occurred in the 17th century. At the beginning of the 17th century the European mathematicians Cavalieri and Fermat were able to find the integrals of polynomial functions of the form f(x)=a1xb1 + a2xb2 + ... for b in the integers (Cavalieri) and in the rationals (Fermat.) While this work arrived at the same results for these functions as the Fundamental Theorem of Calculus would later on, their work failed to achieve the same level of generality that Newton and Leibniz did with their discovery that integration and differentiation are inverses of each other. (Burk 12)

Cauchy, Dirichlet, and Lipschitz

Shortly before the birth of Riemann, the French mathematician Cauchy related the area of the graph beneath any continuous function, differentiable or not, to a sum. Rather than assuming that the antiderivative of the function exists, he broke the domain of integration into subintervals of arbitrary size by a partition (x0, x1, x2, ...), and then calculated the area as the limit of f(x0)(x1 - x0) + f(x1)(x2 - x1) + ... + f(xn)(xn - xn-1) as n increases, and established its existence for any continuous function. (Burk 16) Upon consideration of Fourier's work with discontinuous functions it can be seen that Cauchy's integral was still viable for those discontinuous functions with only a finite number of discontinuities. The integrals could be taken piecewise of all the continuous integrals except for small neighborhoods about the points of discontinuity. (Hawkins 12) So the work of Cauchy lightened the requirements for integrability, from the polynomials of Fermat, to that of the assumed existence of an antiderivative by Newton and Liebniz, to continuuity, and even a finite number of discontinuities. However this trend would continue even further, relaxing the continuity requirement much more.

Fourier series are similar to Taylor series in that they approximate a function by an infinite sum. However, instead of using a power series, a series of sinusoidal terms of the form cisin(bix) is used. This has important applications in physics and signal processing, where it is useful to consider what frequencies are present in a function. In attempting to study the convergence of these series, Dirichlet had occasion to work with functions that may have an infinite number of discontinuities on an interval. An example which will be revisited later is the function f(x) where f(x)=0 when x is ration and 1 when x is irrational. He claimed that these functions would be integrable as long as they were monotonic about any [x - e, x + e] for some e>0 and the set of points of discontinuity was nowhere dense on the domain of integration. This means that there is no interval where for any two points in the set there is a third between them. He claimed that the details would "be presented in another note". However, his promised note never surfaced so the content of his thought will most probably never be known. Due to his inclusion of the continuity requirement, it is unlikely he had in mind the same ideas for integrating these functions as Riemann later had. Instead he probably meant to generalize the integration of a function with a finite number of discontinuities to apply to a function with an infinite number. (Hawkins 14)

Lipschitz's work, although occurring after Riemann's, was most likely not influenced by him, as Riemann's inaugural lecture on integration was not published until after Riemann's death in 1867. This work can better be viewed as an extension to Dirichlet's theories, whereas Riemann branched off in another direction. Lipschitz considered the conditions under which Dirichlet's conditions would not be satisfied. He attempted to prove that if the number of limit points D' of the discontinuities of f(x) is finite, then it is integrable. While he did manage to prove this by taking the integral of the function in the manner of Cauchy everywhere but in arbitrarily small intervals about the points of D', this was in fact a weaker result than Dirichlet's claim. Due to the infancy of set theory, Lipschitz failed to notice the possibility of sets that are infinite but nowhere dense. He did however succeed in removing the requirement of monotonicity and replacing it with the Lipschitz condition. (Hawkins 15) This requires the slopes of all line segments between any two points of the function to be bounded. (Bartle 164)


Riemann was a student of Dirichlet in Berlin, and was influenced by his ideas. He took a more modern view of the function concept from Dirichlet, viewing a function as any correspondence between two sets, not necessarily requiring that it be analytic. In his post-doctoral work Riemann investigated the representation of functions by trigonometric series. This would cover the Fourier series mentioned earlier as well as more complex examples. Dirichlet had covered some of this ground earlier and shared his ideas with Riemann, who said it was reasonable to assume "that the functions not covered by Dirichlet's analysis do not occur in nature." However, he thought some of these yet to be investigated functions might be interesting and applicable in the context of pure mathematics. (Hawkins 17)

He began his investigation by considering what functions are integrable. Riemann's formulation of the integral was quite similar to Cauchy's. Begin by picking an arbitrary set of points x0, x1, ... in ascending order. Then the integral will be the sum of the distance between each consecutive pair of points times the value of the function at some point x'0 somewhere between it. Cauchy picked this point to be in the middle of [xi, xi+1]. Riemann picked it be an arbitrary point on that interval. In studying what classes of functions would be integrable under this definition, Riemann determined that it is necessary and sufficient that f is bounded and the following condition holds. For any e and d positive there is a number c positive so that satisfies the following. Whenever a partition P has a largest interval of size less than c, then the total of all intervals where the difference of the sup and inf of f is greater than d is less than e. Basically, large oscillations of f on the domain of integration must be restricted to a "small" size of intervals. (Burk 17)

Riemann gives an example of a function that is integrable in his sense, and yet is discontinuous at every rational number: f(x)= sigma((nx)/n2) where (nx) is the fractional part of nx, or nx-[nx]. Let D(x) denote the difference between the left and right limits of f(x) where x is a point of discontinuity of f. Analysis of the function reveals that there is a finite number, say c, of discontinuities where D(X)=e, one quarter of that where D(x)=2e and in general c/n2 where D(x)=ne. Since the sum of 1/n2 converges, the total sum of D(x) is finite and hence the function is Riemann integrable. (Burk 19)

Borel, Jordan, Peano

Cantor's initial work in set theory together with Riemann's work on integration led to further extensions of integration theory by Peano, Jordan, and Borel. In 1887 Peano linked the notion of integrability with the "area" of a set S = { (x,y) : aai(S) as the least upper bound of the areas of all polygons contained in S, and the outer area ao(S) as the greatest lower bound of the areas of all polygons that contain S. When these two quantities were equal, the set had that value as its area. (Burk 20) Interestingly this is quite similar in concept to Eudoxus' method of measuring area.

Around five years later, in 1892, Jordan began by dividing the interval into disjoint sets for which a "content" existed. His upper and lower integrals were the sum of the supremums and infimums respectively of the function on each set times the content of the set. When these integrals were equal, the function was Jordan integrable. However, there was a problem with the notion of content used in this way. The contents of the rationals and irrationals on [0,1] are both 1, as is the content of the whole interval [0,1]. For this integral to work well, however, it is necessary for the measure of the union of two intervals to be the sum of the measures of each. (Burk 21)

Borel did no significant direct work on integrals, but his work on laying a groundwork for the measure of sets in 1898 was instrumental to the later work of Lebesgue. His requirements for a measure are as follows (Burk 21):

  1. A measure is nonnegative
  2. The measure of a sum of disjoint sets is the sum of the measure of the individual sets
  3. The measure of the difference of two sets when one is a subset of the other is the difference of the measures
  4. Every set whose measure is not zero is uncountable.


At the end of the 19th century, Lebesgue was completing his studies at Ecole Normale Superieure, a teacher's university in Paris founded in 1794 by the Committee for Public Education. (presentation_anglais.html) It was at this time that Borel's papers on the measure of sets were first published. Lebesgue developed a measure of sets which consists of covering the interval to be measured, E, with a number of disjoint intervals. The outer measure mo is then the greatest lower bound of the lengths of these integrals. The inner measure mi is the size of some interval E' of which E is a subset, less the outer measure of E'-E. When these two measures coincide, the set is Lebesgue measurable. There was some dispute between Lebesgue and Borel as to whose class of sets was larger until 1914, when a Russian mathematician, Suslin, discovered a set which was Lebesgue measurable but not Jordan measurable. (Lebesgue 182)

In the development of his integration theory, Lebesgue thought of the novel idea of partitioning the range of a function rather than its domain. He then took the sets Ei of all values of x where f(x) is in [yi, yi+1]. The advantage of this over the Riemann integral is that the Riemann integral relies on f(x') not varying too quickly. Since Lebesgue's integral bounds this value directly by partitioning the range, this is no longer a concern. (Lebesgue 180)

Lebesgue describes the merits of this method as follows. Suppose you had two merchants counting change. This is analogous to a function from the integers to the integers. Now the merchant can just pick up coins one by one and add them to his sum. A more clever merchant might separate the pennies, nickels, dimes, and quarters. After discovering he has a pennies, b nickels, c dimes, and d quarters, he knows he has a total of a + 5b + 10c + 25d cents. (Lebesgue 182)

As an example, let us consider the integral of f(x) from on [0,1]. Let f(x) be defined as 0 on the rationals and 1 on the irrationals. This function is discontinuous everywhere and has no antiderivative and is not integrable in Riemann's sense. However, we can find the measure of rationals and irrationals on [0,1] quite easily. Since the rationals are countable, their measure must be 0. Any countable set will have measure 0 since it is composed of discrete points, which themselves have measure 0. Since the sum of the measures of two intervals is the measure of the sum of two intervals we can find the measure of the irrationals on [0,1] to be 1 since the measure of all of [0,1] is 1. So the value of this integral is simply 1.


In closing, the work of William Young should also be mentioned. Although he did develop theories of integration and measure as Lebesgue did, he never connected them together. He was however, responsible for interpreting Lebesgue's integral with upper and lower sums as an extension and generalization of Jordan's integral. (Burk 22)

Overall, though Riemann and Lebesgue are seen as the most important figures in recent developments in the theory of integration, it was in reality a much more gradual process. There were many small advances made by various talented mathematicians. However, Riemann and Lebesgue were the last members of two schools of thought. Riemann was the last to make significant progress with considering the integral on intervals rather than on sets. Lebesgue was the last to make progress from a set theoretic approach and place it in an easily usable form. This would explain why the contributions of other's tend to be lost in the background.