The centenary of the Lebesgue integral (Q2708177)

One century ago two major related notions invaded mathematical analysis: the concept of measure [\textit{É. Borel}, Leçons sur la theorie des fonctions. Paris, Gauthier-Villars (1898; JFM 29.0336.01)] and a new concept of integration [\textit{H. Lebesgue}, Leçons sur l'intégration et le recherche des fonctions primitives, 2nd ed. Paris, Gauthier-Villars (1950)].NEWLINENEWLINENEWLINEBorel's statement was sharp, concise, but lacked evidence of existence and unicity of his measure. Lebesgue's revolutionary ideas may be found in a note to the Paris Academy of Sciences. The article under review is related to the centennial commemoration of Lebesgue's communication, dated April 29, 1901, the very `first mathematical achievement of the XXth century'.NEWLINENEWLINENEWLINEAlong 13 lines of the two and a half pages article Lebesgue insists that his switch from the `\(x\)-axis' to the `\(y\)-axis' does not alter Riemannian integration for continuous monotonous functions. He still refrains from considering unbounded functions.NEWLINENEWLINENEWLINEThe authors of this centennial publication reproduce the original article and add important comments. They acknowledge that Lebesgue introduced the definition of (Lebesgue) measurable sets carefully and recall that the break-through of Lebesgue's theory became apparent in the works of Fischer and F. Riesz: the completeness of the space \(L^p\) (the symbolic letter \(L\) was introduced by Riesz).NEWLINENEWLINENEWLINEThe authors mention well-known landmarks in mathematical analysis during the XXth century that may be considered as belonging more or less directly to the Lebesgue heritage.