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Autour Du Centenaire Lebesgue

Gustave Choquet, Thierry De Pauw, Pierre de la Harpe, Jean-Pierre Kahane, Hervé Pajot, and Bruno Sévennec
Société Mathématique de France; distributed by the AMS
Publication Date: 
Number of Pages: 
Panoramas et Synthèses 18
[Reviewed by
Fernando Q. Gouvêa
, on
Since 2001 marked the centennial of the publication of Lebesgue's original note describing his integral, a celebratory conference was held at the École Normale Superieure de Lyon. This book is the result, and it's definitely worth looking at.

The first two essays in the book are historical. Gustave Choquet describes the careers and thought of Borel, Baire, and Lebesgue, who were working on similar topics at about the same time. Jean-Pierre Kahane tells the story of the impact of Lebesgue's integral on 20th century mathematics, providing a link between Choquet's article and those that follow.

The last four essays are expository accounts of more recent work. Pierre de la Harpe writes about the Banach-Tarski paradox and related results, then moves on to finitely additive measures. Bruno Sévennec covers invariant measures on compact groups and results about equidistribution. Thierry de Pauw gives us a tour of conditionally convergent integrals (such as the Henstock integral) and relates them to generalizations of the divergence theorem. Hervé Pajot talks about rectifiability and the "geometric traveling salesman problem".

The first two (maybe even the first three) chapters will be of interest to a wide range of mathematicians, particularly those interested in the history of the calculus. This centennial celebration is one in which I am glad to take part, even if passively.

The table of contents is not available.