April 21, 1997

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The Theory of
Riemann Integration

Through the work on calculus, particularly integration, and its application throughout the century was formidable, there was no actual ``theory'' for it.

The applications of calculus to problems of physics, i.e. partial differential equations, and the fledgling ideas of function representation by trigonometric series required clarification of just what a function was. Correspondingly, this challenged the notion that an integral is just an antiderivative.

Let's trace this development of the integral as a rough and ready way to solve problems of physics to a full-fledged theory.

1. Leonhard Euler (1707-1783) and Jean d'Alembert (1717-1783) argue in 1730-1750's over the ``type'' of solutions that should be admitted as solutions to the wave equation

D'Alembert showed that a solution must have the form

For t=0 we have the initial shape f(x).

Note: Here a function is just that. The new notation and designation are fixed.

But just what kinds of functions f can be admitted?

2. D'Alembert argued f must be ``continuous'', i.e. given by a single equation. Euler argued the restriction to be unnecessary and that f could be ``discontinuous'', i.e. it could be formed of many curves.

In the modern sense though both are continuous.

3. Daniel Bernoulli (1700-1782) entered the fray by announcing that solutions must be expressible in a series of the form

where L is the length of the string.

Euler, d'Alembert and Joseph Lagrange (1736-1813) strongly reject this.

4. In the 19 century the notion of arbitrary function again took center stage when Joseph Fourier (1768-1830) presented his celebrated paper on heat conduction to the Paris Academy (1807). In its most general form, Fourier's proposition states:

Any (bounded) function f defined on (-a,a) can be expressed as

where

5. For Fourier the notion of function was rooted in the 18 century. In spite of the generality of his statements a ``general" function for him was still continuous in the modern sense. For example, he would call

discontinuous.

6. Fourier believed that arbitrary functions behaved very well, that any f(x) must have the form

which is of course meaningless.

7. For Fourier, a general function was one whose graph is smooth except for a finite number of exceptional points.

8. Fourier believed and attempted to validate that if the coefficients could be determined then the representation must be valid.

His original proof involved a power series representation and some manipulations with an infinite system of equations.

Lagrange improved things using a more modern appearing argument:

(a) multiply by , (b) integrate between -a and a, term-by-term, (c) interchange to . With the ``orthogonality" of the trig functions the Fourier coefficients are achieved.

The interchange to was not challenged until 1826 by Niels Henrik Abel (1802-1829).

The validity of term-by-term integration was lacking until until Cauchy proved conditions for it to hold.

ß

Nonetheless, even granting the Lagrange program, the points were still thought to be lacking validity until Henri Lebesgue (1875-1941) gave a proper definition of area from which these issues are simple consequences.

9. Gradually, the integral becomes area based rather than antiderivative based. Thus area is again geometrically oriented. Remember though......... ß

Area is not yet properly defined.ß

And this issue is to become central to the concept of integral.

10. It is Augustin Cauchy (1789-1857) who gave us the modern definition of continuity and defined the definite integral as a limit of a sum. He began this work in 1814.

11. In his Cours d'analyse (1821) he gives the modern definition of continuity at a point (but uses it over an interval). Two years later he defines the limit of the Cauchy sum

as the definite integral for a continuous function. Moreover, he showed that for any two partitions, the sums could be made arbitrarily small provided the norms of the partitions are sufficiently small. By taking the limit, Cauchy obtains the definite integral.

The basic refinement argument is this: For continuous (i.e. uniformily continuous) functions, the difference of sums and can be made arbitrarily small as a function of the maximum of the norms of the partitions.

This allowed Cauchy to consider primitive functions,

He proved:

• Theorem I. F is a primitive function; that is F'=f
• Theorem II. All primitive functions have the form .

To prove Theorem II he required

• Theorem III. If G is a function such that G'(x)=0 for all x in [a,b], then G(x) remains constant there.

Theorems I, II and III form the Fundamental Theorem of Calculus. The proof depends on the then remarkable results about partition refinement. Here he (perhaps unwittingly) envokes uniform continuity.

12. Nonetheless Cauchy still regards functions as equations, that is y=f(x) or f(x,y)=0.

13. Real discontinuous functions finally emerge as those having the form

where is a partition of [a,b] and each is a continuous (18 century) function on . Cauchy's theory works for such functions with suitable adjustments. For this notion, the meaning of Fourier's and is resolved.

14. Peter Gustav Lejeune-Dirichlet (1805-1859) was the first mathematician to call attention to the existence of functions discontinuous at an infinite number of points. He gave the first rigorous proof of convergence of Fourier series under general conditions by considering partial sums

and showing

(Is there a hint of Vieta here?)

In his proof, he assumes a finite number of discontinuities (Cauchy sense). He obtains convergence to the midpoint of jumps. He needed the continuity to gain the existence of the integral. His proof requires a monotonicity of f.

15. He believed his proof would adapt to an infinite number of discontinuities; which in modern terms would be no where dense. He promised the proof but it never came. Had he thought of extending Cauchy's integral as Riemann would do, his monotonicity condition would suffice.

16. In 1864 Rudolf Lipschitz (1831-1904) attempted to extend Dirichlet's analysis. He noted that an expanded notion of integral was needed. He also believed that the nowhere dense set had only a finite set of limit points. (There was no set theory at this time.) He replaced the monotonicity condition with piecewise monotonicity and what is now called a Lipschitz condition.

Recall, a function f(x), defined on some interval [a,b] is said to satisfy a Lipschitz condition of order if for every x and y in [a,b]

for some fixed constant c. Of course, Lipschitz was considering .

Every function with a bounded derivative on an interval, J, satisfies a Lipschitz condition of order 1 on that interval. Simply take

17. In fact Dirichlet's analysis carries over to the case when is finite (D= set, D':= limit points of D), and by induction to . (Such sets were introduced by George Cantor (1845-1918) in 1872.)

Example. Consider the set . Then .ß

Example. Consider the set , of all rationals. Then , where R is the set of reals.

Dirichlet may have thought for his set of discontinuities is finite for some n. From Dirichlet we have the beginnings of the distinction between continuous function and integrable function.

18. Dirichlet introduced the salt-pepper function in 1829 as an example of a function defined neither by equation nor drawn curve.

Note. Riemann's integral cannot handle this function. To integrate this function we require the Lebesgue integral.

By way of background, another question was raging during the 19th century, that of continuity vs. differentiability. As late as 1806, the great mathematician A-M Ampere (1775-1836) tried without success to establish the differentiability of an arbitrary function except at ``particular and isolated" values of the variable.

In fact, progress on this front did not advance during the most of the century until in 1875 P. DuBois-Reymond (1831-1889) gave the first conterexample of a continuous function without a derivative.

The Riemann Integral

Bernhard Riemann (1826-66) no doubt acquired his interest in problems connected with trigonometric series through contact with Dirichlet when he spent a year in Berlin. He almost certainly attended Dirichlet's lectures.

For his Habilitationsschrift (1854) Riemann under-took to study the representation of functions by trigonometric functions.

He concluded that continuous functions are represented by Fourier series. He also concluded that functions not covered by Dirichlet do not exist in nature. But there were new applications of trigonometric series to number theory and other places in pure mathematics. This provided impetus to pursue these foundational questions.

Riemann began with the question: when is a function integrable? By that he meant, when do the Cauchy sums converge?

He assumed this to be the case if and only if

where P is a partition of [a,b] with the lengths of the subintervals and the are the corresponding oscillations of f(x):

For a given partition P and , define

Riemann proved that the following is a necessary and sufficient condition for integrability (R2): Corresponding to every pair of positive numbers
and there is a positive d such that
if P is any partition with norm
, then S(P,)<.

These conditions and are germs of the idea of Jordan measurability and outer content. But the time was not yet ready for measure theory.

Thus, with and Riemann has integrability without explicit continuity conditions. Yet it can be proved that R-integrability implies f(x) is continuous almost everywhere.

Riemann gives this example: Define m(x) to be the integer that minimizes |x-m(x)|. Let

(x) is discontinuous at x=n/2 when n is odd. Now define

This series converges and f(x) is discontinuous at every point of the form x=m/2n, where (m,n) = 1. This is a dense set. At such points the left and right limiting values of this function are

This function satisfies and thus f is R-integrable.

The R-integral lacks important properties for limits of sequences and series of functions. The basic theorem for the limit of integrals is:

Theorem. Let J be a closed interval [a,b], and let be a sequence of functions such that

and such that tends uniformily to f(x) in J as . Then

That this is unsatisfactory is easily seen from an example. Consider the sequence of functions defined on [0,1] by . Clearly, as , pointwise on [0,1) and , for all n. Because the convergence is not uniform, we cannot conclude from the above theorem that

which, of course, it is.

What is needed is something stronger. Specifically if and , g are integrable and if then f may not be R-integrable.

This is a basic flaw that was finally resolved with Lebesgue integration.

Postscript.

The (incomplete) theory of trigonometric series, particularly the question of representability, continued to drive analysis. The most difficult question was this: what functions are Riemann integrable? To this one and the many other questions that arose we owe the foundations of set theory and transfinite induction as proposed by Georg Cantor. Cantor also sought conditions for convergence and defined the derived sets . He happened on sets

and so on, which formed the basis of his transfinite sets. Another aspect was the development of function spaces and ultimately the functional analysis that was needed to understand them. In a not uncommon reversal we see so much in mathematics, these spaces have played a major role in the analysis of solutions of the partial differentials equations and trigonometric series that initiated their invention. Some of the most active research areas today are direct decendents of this question of integrability.

I might add that these pursuits were fully in concordance with the fundamental philosophy laid down by the Pythagorean school more than two millenia ago.

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