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.
Highlights
In (Biacino 2018) the evolution of the concept of a real function at the beginning of the 20thcentury, after the introduction in the mathematical world of the Dirichlet’s concept of function, the appearance of a series of pathological functions and the definitions of new classes of sets and functions given by the French mathematicians Baire (1874-1932), Borel (1871-1956) and Lebesgue (1875-1941), is examined
Dini (1845-1918); this is why the theses of Baire, in 1898, and Lebesgue in 1902, were published on the Annali di Matematica, directed by Dini; the important mathematician had already developed in his famous treatise: Fondamenti per la teorica delle funzioni di variabili reali in 1878 noteworthy attempts at the study of the functions of a real variable in a very general setting; there he gives the notion of indetermination limits in a point, on the left and on the right, for oscillating functions and the consequent definition of derived numbers, the so called Dini derivatives, the proof of the Hankel condensation principle and a great amount of fundamental theorems, we will often quote in this paper
After the absolutely continuous additive set functions, Lebesgue studies in his paper Sur l’intégration des fonctions discontinues the functions of bounded variation of more than one variable in an analogous way
Summary
In (Biacino 2018) the evolution of the concept of a real function at the beginning of the 20thcentury, after the introduction in the mathematical world of the Dirichlet’s concept of function, the appearance of a series of pathological functions and the definitions of new classes of sets and functions given by the French mathematicians Baire (1874-1932), Borel (1871-1956) and Lebesgue (1875-1941), is examined. The proof, we give the only if part of it, is very simple because, even if it requires more advanced notions, it is not linked to du Bois-Reymond’s criterion, but is related directly to the Riemann definition of integrability On this occasion, Vitali makes use of the following property: given a real, bounded function in an interval (a,b), the upper The very simple proof by Lusin is similar to the preceding one given by Lebesgue: it is based on the fact that if f is of class 1, by Severini Egoroff theorem, the following property holds: For every >0 there exists a closed set P such that meas(P)>1- and f is continuous on P. He defines the sets of null measure as usual, for every set he calls it measurable if it is the union of a closed set and of a set of null measure: he does not give a definition of measure but he can define in this way measurable functions (Sierpinski 1916)
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