Contribution of Italian Mathematicians to Real Analysis in the last Decades of Nineteenth Century

Loredana Biacino

doi:10.24117/2526-2270.2019.i7.02

Abstract

In (Biacino 2018) the evolution of the concept of real function of a real variable at the beginning of 1900 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, I consider in particular the contribution to real analysis theory done by some Italian mathematicians as Volterra, Peano, Ascoli, Arzelà, etc., in the last decades of nineteenth century before the introduction of measure and integration theory by Lebesgue.

Highlights

One of the most interesting problems many mathematicians were faced in the second half of nineteenth century can be outlined as follows: there are two ways to define the integral, in order to calculate an area
Riemann had met in Berlin the Italian mathematicians Betti, Brioschi and Casorati in 1858 and had established friendly relation with Betti and Beltrami, at that time professors at Pisa, living for one year in that town in 1864
Arzelà proves something similar to the theorem that if a sequence of measurable functions almost everywhere converges in a bounded interval (a, b) it converges in measure in (a, b), even if Arzelà did not use a general concept of measure yet. Perhaps he was inspired by the fact that the requirement of something similar to convergence in measure, with respect to outer content, had appeared briefly for the first time in 1878 in a paper about the problem of term-by-term integration by Kronecker (Hawkins 2002, 111-2)

Summary

Introduction

One of the most interesting problems many mathematicians were faced in the second half of nineteenth century can be outlined as follows: there are two ways to define the integral, in order to calculate an area. With regard to Hankel theory of pointwise discontinuous functions Dini pointed out that if a function f is Riemann integrable it is pointwise discontinuous (Dini 1878, 250) that is it has infinitely many discontinuities but in every subinterval of the interval where it is defined there is at least a point of continuity for f (Dini 1878,62); but he claimed that he believed that the converse in general does not hold He was not able to construct an example showing that there exist nowhere dense sets whose content is positive to prove the partial fallacy of Hankel proof. Severini written in 1897 where the Weierstrass approximation theorem is extended to a class of integrable functions

A New Characterization of Integrable Functions by Ascoli and du Bois-Reymond

Conclusion