Giuseppe Vitali: Real and Complex Analysis and Differential Geometry

Vitali’s most significant output took place in the first eight years of the twentieth century when Lebesgue’s measure and integration were revolutionising the principles of the theory of functions of real variables. This period saw the emergence of some of his most important general and profound results in that field: the theorem on discontinuity points of Riemann integrable functions (1903), the theorem of the quasi-continuity of measurable functions (1905), the first example of a non-measurable set for Lebesgue measure (1905), the characterisation of absolutely continuous functions as antidervatives of Lebesgue integrable functions (1905), the covering theorem (1908). In the complex field, Vitali managed to establish fundamental topological properties for the functional spaces of holomorphic functions, among which the precompacity of a family of holomorphic functions (1903-04). Vitali’s academic career was interrupted by his employment in the secondary school and by his political and trade union commitments in the National Federation of Teachers of Mathematics (Federazione Nazionale Insegnanti di Matematica, FNISM), which brought about a reduction, and eventually a pause, in his publications. Vitali took up his research work again with renewed vigour during the national competition for university chairs and then during his academic activity firstly at the University of Modena, then Padua and finally Bologna. In this second period, besides significant improvements to his research of the first years, his mathematical output focussed on the field of differential geometry, a discipline which in Italy was long renowned for its studies, and particularly on some leading sectors like connection spaces, absolute calculus and parallelism, projective differential geometry, and geometry of the Hilbertian space. Giuseppe Vitali’s mathematical output has been analysed from various points of view: his contributions to real analysis, celebrated for their importance in the development of the discipline, were accompanied by a more correct evaluation of his works of complex analysis and differential geometry, which required greater historical investigation since the language and themes of those research works underwent various successive developments, whereas the works on real analysis maintained a collocation within the classical exposition of that theory. This article explores the figure of Vitali and his mathematical research through the aforementioned contributions and, in particular, the edition of memoirs and correspondences promoted by the Unione Matematica Italiana, which initiated and encouraged the analysis of his scientific biography.