Development of the Theory of the Functions of Real Variables in the First Decades of the Twentieth Century

Abstract

In (Biacino 2018) the evolution of the concept of a real function of a real variable at the beginning of the twentieth century is outlined, reporting the discussions and the polemics, in which some young French mathematicians of those years as Baire, Borel and Lebesgue were involved, about what had to be considered a genuine real function. In this paper a technical survey of the arising function and measure theory is given with a particular regard to the contribution of the Italian mathematicians Vitali, Beppo Levi, Fubini, Severini, Tonelli etc … and also with the purpose of exposing the intermediate steps before the final formulation of Radom-Nicodym-Lebesgue Theorem and the Italian method of calculus of variations.