Autour du centenaire Lebesgue
On April 29, 1901, Henri Lebesgue published a note in Comptes-Rendus de l’Académie des Sciences with the title Sur une généralisation de l’intégrale définie. On the occasion of the centenary of the birth of the Lebesgue integral, the conference “La mesure de Lebesgue a 100 ans” was organized by l’Ecole Normale Supérieure de Lyons. The book under review results from this celebration and provides various viewpoints on Lebesgue’s heritage in an excellent way.
The contribution by G. Choquet provides a vivid testimony to mathematics and mathematicians of Lebesgue’s era including E. Borel and R. Baire. It mentions their predecessors, their personalities, the sources of their inspiration and their mutual relations. P. de la Harpe presents a nice exposition of results on paradoxical decompositions and invariant finitely additive measures. The importance of this subject for group theory is emphasized. Classical as well as contemporary results are analyzed and at the end of the paper, several open problems on amenability for groups are proposed. B. Sévennec’s article surveys some equidistributive results in compact groups. The beautiful von Neumann’s proof of existence and uniqueness of Haar’s measure is presented as an illustration on how the measure may be obtained as a limit of “equidistributed” measures with finite support. The “spectral gap” of averaging operators is discussed and recent applications are given (e.g. to Ruziewicz’s problem from 1916 on uniqueness of invariant finitely additive invariant measure on the Lebesgue measurable sets). T. De Pauw’s paper is devoted to the divergence theorem from the point of view of non-absolutely convergent integrals. It is a nice survey on Henstock-Kurzweil integration, sets of finite perimeter and W.F. Pfeffer’s approach to the divergence formula. The contribution by H. Pajot presents the recent progress on understanding the notion of rectifiability in relation to the analytic capacity or the Cauchy operator. At about 150 pages, the book yields rich material on very attractive subjects. All papers have a high level of exposition, are well organized and readable. A preface by J.-P. Kahane highlights Lebesgue’s influence in the course of 20th century. It presents a nice synthetic viewpoint on the history and the development of mathematics during the last hundred years. This extremely interesting book can be recommended to anybody who likes mathematics.