Your son Giuseppe, who has been my student for five years in the classical school of Ravenna, has always shown great attitude and lots of love for the study of mathematics, so I harbour the highest hopes for him, if he may be given the opportunity to continue to higher studies. His marks, combined with his goodness and modesty, make him the best student I have had so far.

At that time it was not unusual for a person to begin a university career with a period of secondary school teaching (other examples are Luigi Cremona and Cesare Arzelà), but it was a great pity that it took the university up till 1922 to concede full research activity to such talent like that of Vitali. Vitali, however, kept in contact with his mentors, and Arzelà in particular remained a constant point of reference for him, through exchange of letters and also meetings above all during the holiday periods (Vitali returned to Bologna during his teaching years in Voghera, and Arzelà, in his turn, spent long periods in Santo Stefano Magra in Liguria). He always maintained a friendship with Enriques, who had followed his transfer from the University of Bologna to the Scuola Normale Superiore of Pisa, but his lifelong friend was Guido Fubini, who had been a fellow student in Pisa.

The works of [Vitali] show his through understanding of the various branches of analysis, his acumen in the treatment of questions that are delicate and not easy, arriving at interesting results, remarkable clarity and sobriety of composition. The fact itself of his having met with one of the creators of the new directions of integral calculus, Lebesgue, is proof that he stayed within the mainstream of this research, in which such results naturally presented themselves.

To the young people who dedicate themselves to the study of mathematics, I say that the ability required to find the most suitable and most elegant devices is acquired through practise, through the examination of many examples, through the effort to imitate. To imitate, but not too much, if our discipline is not to become a marsh, a large one to be sure, but stagnant, with neither life nor movement. First imitate to learn, and then renew ourselves. ... I love, at least when I am able, to regard science from a personal point of view, and always, again when possible, go beyond current opinions and look at the problem from a new perspective. I have the impression that some ways must be left behind, some mental habits must be abandoned, if we are not to clip the wings of progress. Even to science we must sometimes repeat Charon's cry: By another way, by other ports, not here, you will find passage across the shore. In my role as teacher I hope to be able to show you other ways, if not other ports.

I had just graduated a few months before in pure mathematics, and was a volunteer assistant in the course of the theory of functions. Vitali was at the height of his scientific career, both in terms of the recognition he received - alas, so tardily! - by the highest Italian academies, and for having begun and, in part, concluded the preparation of his treatises, as well as for having begun, with renewed, surprising creative capacities, that research in stellar astronomy ... Vitali was extraordinarily generous and good to me. Beginning with my degree thesis, at that time I was publishing my first works on functions of a real variable with a unilateral derivative. He came to see me, spoke with me at length, he presented me with extracts of his most beautiful, most famous publications on set theory, and on functions of a real variable, he gave me much good advice, he made it so I opened up to him about the whole of my studies and my projects for the future, and that I kept him informed about the results - alas, so modest! - of my research.

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 nonmeasurable 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 theorem of compacity of a family of holomorphic functions (1903-1904).

In a recent work, Professor Beppo Levi introduces for didactic purposes a new definition of integral for bounded functions. He writes: "For measurable functions the notion of integral we introduce coincides with the Lebesgue integral: however, it might be the case that this notion may be applied to nonmeasurable functions". I prove that the definition of Beppo Levi is actually altogether equivalent to that of Lebesgue.

The impression on my mind made by the announcement of his death can never be erased, an announcement that reached me in Paris, during a seminar I was taking part in. I heard people whispering "Vitali est mort!" A few days later I received the painful confirmation directly from Bologna: the Maestro had fallen, struck down suddenly while walking arm in arm with his colleague Ettore Bortolotti, under the porticos of that learned city in which I had spent the most beautiful years of the university studies, and in which were laid, and lie still, the remains of those who gave me life.