In 1980, I was at something of a career impasse. I had done a Ph.D. in topology and was teaching at a small liberal arts college, but classroom responsibilities and endless committee meetings threatened to extinguish my mathematical flame. I needed some kind of scholarly recharge.
At just that moment, I picked up Lebesgue’s Theory of Integration by Thomas Hawkins. The book was like nothing I had ever read. Hawkins’s account of 19th century analysis is a tale of failure and success, of gaffes and insights, of mathematicians great and small who paved the way for the incomparable Henri Lebesgue and his wonderful integral. It is a mathematical adventure story.
First, Hawkins sets the stage. Through the 18th century, integration had been regarded primarily as anti-differentiation, a tool for solving the multitude of differential equations that had so enriched the sciences. This left the integral as a secondary concept, lost in the shadow of its more significant cousin, the derivative.
In the 19th century, this changed. Through the work of Cauchy in 1823 and, especially, of Riemann in 1854, the integral came into its own. Domains were partitioned; sums were formed; and the integral was thereby defined through a limiting process. For the first time, integrability assumed its rightful place alongside continuity and differentiability as a central analytic concept. Indeed, it would soon become primus inter pares.
Riemann stood triumphant. As Hawkins notes, mathematicians generally believed that “Riemann had extended the concept of an integrable function to its outermost limits.” As a case in point, this definition allowed for highly discontinuous functions to be integrated — Riemann himself introduced an integrable function that, although continuous on a dense set, was discontinuous on a dense set as well.
Yet mathematicians knew that a function could not be too discontinuous if it was to be integrated. After all, in 1829 Dirichlet had defined his nowhere continuous function (equal to 1 on the rationals and 0 on the irrationals) that failed to be integrable under Riemann’s definition.
Thus, the continuity/integrability relationship needed to be clarified. To do so, analysts sought to finish this critical sentence: “A function is Riemann-integrable if and only if it is continuous except at ____________,” where the blank was to be filled by some kind of “small” set.
Hawkins describes how attempts to fill this blank went astray, hampered by a dearth of examples and a poor understanding of set theory. Mathematicians found themselves at dead ends after introducing concepts like “pointwise discontinuity” or “sets in loose order,” concepts that leave modern readers scratching their heads in bewilderment.
Then the strange counterexamples started to appear. For instance, in 1881 Vito Volterra introduced a non-integrable function whose points of discontinuity formed a nowhere dense set. Although “nowhere dense” embodied a kind of topological smallness, it obviously was not the blank-filler.
Things got worse. The same Volterra found an everywhere differentiable function whose (bounded) derivative was — horrors! — not integrable by Riemann’s definition. This effectively killed the fundamental theorem of calculus under Riemann’s theory because, although this function \(f\) and its derivative \(f'\) were perfectly well defined, the equation \[ \int_a^b f'(x)\,dx = f(b)-f(a)\] was untrue, not because the left-hand side didn’t equal the right-hand side but because the left-hand side did not even exist.
Matters had become most untidy. Riemann’s integral had dominated mid-19th century analysis so thoroughly that a viable alternative, in Hawkins’s words, “seemed unthinkable.” Yet counterexamples such as Volterra’s deprived the Riemann integral of its sense of inevitability. Maybe a new approach was in order.
Hawkins now brings us to the brilliant conclusion. Henri Lebesgue, building upon the work of such predecessors as Georg Cantor and Émile Borel, turned analysis on its head. First off, he settled the continuity/integrability question by proving that a function is Riemann integrable if and only if it is continuous except on a set of “measure zero.” This was the smallness property that mathematicians had been seeking for decades.
But Lebesgue was just getting started. He extended the concept of measure to a vast collection of subsets of real numbers and used this extension to re-define the integral by the bold act of partitioning not the domain of a function but its range. In the process, a host of technical details had to be addressed — Hawkins does not shy away from these — but the outcome was the Lebesgue integral, one of the great achievements in the history of analysis. Armed with this new idea, Lebesgue showed that many of the “flaws” of Riemann’s approach fell by the wayside. As he himself put it, “a generalization made not for the vain pleasure of generalizing but in order to solve previously existing problems is always a fruitful [one].”
At this point, two observations are in order. First, the story Hawkins tells is vastly more complicated than is suggested by this brief synopsis. He addresses other thorny issues from the time, such as when/whether the integral of the limit is the limit of the integrals. And he samples the work of mathematicians like Darboux, Thomae, Hankel, Harnack, and Baire. Some of these individuals will be familiar to modern readers and some will not, but each played a role in the tale.
Second, as should be clear from the account above, Hawkins cuts no mathematical corners. His book is technically challenging and is not to be attempted without a solid grounding in real analysis. Be forewarned.
Let me conclude this review by returning to my personal story. When I put down Hawkins’s book all those years ago, I saw how I could recharge my batteries. I would plunge into the history of mathematics, would try to learn the story contained therein, and would hope to share that story with others. Insofar as I have traveled such a road, I owe much of my journey to this excellent book.
And you need not take my word for its excellence. In 2001, Thomas Hawkins received the first Whiteman Prize from the American Mathematical Society for “notable exposition in the history of mathematics.” The accompanying AMS citation began (of course!) by celebrating Lebesgue’s Theory of Integration: Its Origins and Development as a “classic in the field.”
And so it is.
William Dunham is currently Research Associate in Mathematics at Bryn Mawr College. He is the author of many books, and most recently was one of the editors of The G. H. Hardy Reader.