Giuseppe Vitali: Research on Real Analysis and Relationship with Polish and Russian Mathematicians

Giuseppe Vitali (1875-1932) was one of the most interesting protagonists of Italian and European mathematics in the early twentieth century. His most significant production took place in the field of real and complex analysis over a short period: the first decade of the century. It was then that he achieved success in demonstrating some remarkable results, such as the necessary and sufficient condition for Riemann integrability, the first example of non–Lebesgue measurable sets, the theorem of compacity of a family of holomorphic functions, the covering theorem, which are still of great relevance in the literature of mathematics today. His research was linked to the new field opened in France by Lebesgue and Borel, and which in Italy had originated from the work of Ulisse Dini and his school. After a long period of teaching in secondary schools, owing to the difficulty in obtaining an academic position, Vitali returned to university in 1922 and he had to catch up with the latest research: in the meanwhile the new analysis was being developed by mathematicians in Russia such as Egorov and Luzin or in Poland like Sierpiński, Mazurkiewicz, Nikodym and Banach