Federico Cafiero
Federico Cafiero  

Born 
Riposto  24 May 1914
Died 
7 May 1980 65) Napoli  (aged
Nationality  Italian 
Fields  
Institutions  
Alma mater  Università degli Studi di Napoli Federico II 
Doctoral advisor  Renato Caccioppoli 
Doctoral students 

Notable awards 

Federico Cafiero (24 May 1914 – 7 May 1980) was an Italian mathematician known for his contributions in real analysis, measure and integration theory, and in the theory of ordinary differential equations. In particular, generalizing the Vitali convergence theorem, the Fichera convergence theorem and previous results of Vladimir Mikhailovich Dubrovskii, he proved a necessary and sufficient condition for the passage to the limit under the sign of integral:^{[3]} this result is, in some sense, definitive.^{[4]} In the field of ordinary differential equations, he studied existence and uniqueness problems under very general hypotheses for the left member of the given first order equation, developing an important approximation method and proving a fundamental uniqueness theorem.^{[5]}
Life and academic career
Cafiero was born in Riposto, Province of Catania, on May 24, 1914.^{[6]} He obtained his Laurea in mathematics, cum laude, from the University of Naples Federico II in 1939.^{[7]} During the 1939–1940 academic year, he won an "Istituto Nazionale di Alta Matematica" scholarship and went in Rome to the institute:^{[8]} there he followed the courses held by Francesco Severi, Mauro Picone, Luigi Fantappiè, Giulio Krall and Leonida Tonelli.^{[9]}
The World War II years: 1941–1943
He was appointed instructor of the course of "Elementi di matematica"^{[10]} by the Faculty of Statistical Sciences of the University of Rome, for the 1940–1941 academic year:^{[11]} however, he was able to hold the course only for few months, since he was called to arms on January 1941^{[12]} and stationed from May 1942 to September 1943 on the Northern African coasts as an officer of the San Marco Battalion.^{[13]} It was there that, after having successfully completed a dangerous sabotage operation, the Armistice between Italy and Allied armed forces surprised him and the other members of his unit, leaving them without any support.^{[12]} Nonetheless, in desperate conditions, he was able to lead his men to the Italian coasts with a rubber dinghy, and was awarded of a Silver Medal of Military Valor for this act.^{[12]}
Rebuilding and researching: the years from 1944 to 1953
Being discharged from Military Service on February 1944,^{[7]} he was not able to reach Rome and remained in Napoli.^{[12]} The institution which currently is the Institute of Mathematics of the University of Naples was on the way of reconstituting,^{[14]} the eight former mathematics institutes of the university having been literally "torn to pieces" by the Allied forces Military Police.^{[15]} It was necessary to collect and reorder in a new library all the volumes of the previously existed ones, then piled on the floor of a single room, catalogue them exnovo and create new records, provide the library administration, and of course there was no administrative personnel available nor financial resources.^{[16]} It was also necessary to organize courses and exams for the numerous war veterans coming back from the front and for new students, with more than a half of the teaching personnel blocked beyond the Gothic Line:^{[15]} and in performing all those task Cafiero, jointly with few others and working as an adjuct professor of "Esercitazioni di Matematiche", was an outstanding collaborator of Renato Caccioppoli and Carlo Miranda.^{[17]}
Also in 1944 he married Jole Giorgini, his lifelong companion, and soon after they had a daughter, Anna.^{[7]}
Due to the scarce possibilities of being hired permanently by the Faculty of Sciences at that time, he accepted a position as adjunct assistant professor to the chair of Financial Mathematics,^{[18]} working with Luigi Lordi first at the Istituto Universitario Navale and then to the Faculty of Economics and Business, where he was appointed full assistant professor in June 1949.^{[19]} Nonetheless, his ties with the Faculty of Sciences remained strong, being employed there as adjunct professor of "Esercitazioni di Matematiche" several times, during those years:^{[20]} he was likewise assigned to several other courses related to Financial Mathematics by the Istituto Universitario Navale and by the Faculty of Business and Economics.^{[20]}^{[21]}
The scientific aspect of the collaboration with the Faculty of Sciences was nonetheless very intense,^{[20]} leading him to the "libera docenza" in March 1951, and to a full professorship chair in 1953:^{[22]}^{[23]} during this period, his scientific activity was done side by side at first with Carlo Miranda and later with Renato Caccioppoli, who found in him a dear pupil and friend.^{[24]}
Ranked first of the three winners of the competition for the chair of mathematical analysis of the University of Catania,^{[25]} on December 1953 he was appointed as extraordinary professor to that chair, and left Napoli for Catania.^{[26]}^{[27]}
First in Catania and then in Pisa: the years from 1954 to 1959
Cafiero started his service at the University of Catania on January 1954.^{[28]} His arrival at the university brought several innovations, both in teaching and in the research activity on mathematical analysis.^{[27]}^{[29]} In particular, he established a seminar on abstract measure theory open to assistant professors and to graduate students as well, and this was felt as true scientific revolution:^{[29]} he held the chair of mathematical analysis for three years.^{[30]} After becoming ordinary professor in 1956,^{[7]} he went to the University of Pisa on request by Sandro Faedo:^{[31]} during his stay, he held courses also at the Scuola Normale Superiore,^{[32]} becoming director of the "Leonida Tonelli" Institute and member of Board of directors of the Centro Studi Calcolatrici Elettroniche.^{[33]}
Work
Research activity
Ma è subito dopo la seconda guerra mondiale che il processo di astrattizzazione della teoria della misura e dell'integrale si completa in modo definitivo. A ciò contribuirono Paul Halmos negli U.S.A. e Renato Caccioppoli, Federico Cafiero (1914–1980) ed altri in Italia.^{[34]}
Teaching activity
Come Andreotti anche Stampacchia non poté venire subito a Pisa e così io fui felice di avere con me un altro valoroso allievo di Renato Caccioppoli, Federico Cafiero, che restò a Pisa poco tempo, ma vi lasciò una forte traccia e formò il suo valido continuatore Giorgio Letta.^{[35]}
Selected publications
The papers of Federico Cafiero listed in this section are also included in his "Opere scelte" (Cafiero 1996), which collect all his published notes and one of his books.
 Cafiero, Federico (1953), "Sul passaggio al limite sotto il segno d'integrale per successioni d'integrali di StieltjesLebesgue negli spazi astratti, con masse variabili con gli integrandi" [On the passage to the limit under the sign of integral for sequences of Stieltjes–Lebesgue integrals in abstract spaces], Rendiconti del Seminario Matematico della Università di Padova (in Italian), 22: 223–245, MR 57951, Zbl 0052.05003, is the first paper where Federico Cafiero states and proves his convergence theorem.
 Cafiero, Federico (1953a), Funzioni additive d'insieme e integrazione negli spazi astratti [Additive set functions and integration in abstract spaces] (in Italian), Napoli: Libreria Editrice Liguori, p. 178, MR 0215954, Zbl 0050.27801, is the prize winning first monograph where Federico Cafiero states and proves his convergence theorem.
 Cafiero, Federico (1959), Misura e integrazione [Measure and integration], Monografie matematiche del Consiglio Nazionale delle Ricerche (in Italian), 5, Roma: Edizioni Cremonese, pp. VII+451, MR 0215954, Zbl 0171.01503, is a definitive monograph on integration and measure theory: the treatment of the limiting behavior of the integral of various kind of sequences of measurerelated structures (measurable functions, measurable sets, measures and their combinations) is somewhat conclusive.
 Cafiero, Federico (1996), Opere scelte, a cura del Dipartimento di matematica e applicazioni R. Caccioppoli dell'Università degli studi di Napoli Federico II e con il contributo dell'Accademia Pontaniana e dei dipartimenti di matematica delle università di Catania, di Napoli e di Pisa (in Italian), Napoli: Giannini Editore, p. 701. Federico Cafiero's "Selected works", including all his published papers, tree postcards from his master Renato Caccioppoli concerning his research and his book "Lezioni sulla teoria delle funzioni di variabili reali" (English translation: "Lectures on the theory of functions of real variables").
See also
 Carathéodory's existence theorem
 Dominated convergence theorem
 Peano existence theorem
 Picard–Lindelöf theorem
Notes
 ↑ Awarded for the monograph (Cafiero 1953a), according to (De Angelis & Sbordone 1999, p. 29), (de Lucia & Sbordone 1996), (Letta 1981, p. 347) and (Miranda 1980/1981, p. 9). The yearbook of the Accademia Pontaniana (2015, p. 123), in the list of prize recipients, states that the award winning memoir title was:"Studio delle famiglie di funzioni additive di insieme; esposizione sistematica di risultati recenti e nuovi contributi; applicazioni alla teoria generale del passaggio al limite sotto il segno di integrale".
 ↑ See the list of prize winners at the Presidenza della Repubblica Italiana web site.
 ↑ See (Cafiero 1953), (Cafiero 1953a) and (Cafiero 1959, pp. 388–392).
 ↑ According to Letta (1981, pp. 353–354).
 ↑ According to Letta (1981, pp. 349–350), who describes briefly these results, and to Piccinini, Stampacchia & Vidossich (1978), who comprehensively present Cafiero's and other's research results in this field.
 ↑ See (Letta 1981, p. 347) and (Miranda 1980/1981, p. 9): his parents were from Meta di Sorrento, according to Miranda.
 1 2 3 4 See (Letta 1981, p. 347).
 ↑ See (Letta 1981, p. 347), (Miranda 1980/1981, p. 9) and (Roghi 2005, p. 13): Letta and Roghi clearly state the academic year, while Miranda states that he won the scholarship "subito dopo" i.e. "soon after" earning his Laurea degree. Roghi gives many other details about the scholarship, including the names of other winners and its amount, which was 5000 Italian liras.
 ↑ Miranda (1980/1981, p. 9) lists only the first four names while Letta (1981, p. 347) also mentions Tonelli but not Krall. Roghi (2005, p. 13) gives the full list of the courses held at the institute during the 1939–1940 academic year, including the names of the appointed teachers: along the ones cited by Letta and Miranda, Enrico Bompiani, Giovanni Giorgi, Ugo Amaldi, Antonio Signorini and Fabio Conforto are also mentioned.
 ↑ English translation: "Elements of mathematics".
 ↑ According to Letta (1981, p. 347), who reports also that Cafiero was confirmed in the job for the three following years. Miranda (1980/1981, p. 9) presents a slightly different version, referring that that he was appointed instructor of the course of "Esercitazioni di Matematiche" (i.e. "Exercises in mathematics") by the Faculty of sciences. However, the version of Letta has been followed since it is more circumstantial, offering more details.
 1 2 3 4 See (Letta 1981, p. 347) and (Miranda 1980/1981, p. 9).
 ↑ See (Letta 1981, p. 347) and (Miranda 1980/1981, p. 9): unlike the former one, this last source does not state the duration of Cafiero's stay in Africa.
 ↑ Description of the state the Institute at the time, as reported here, is taken from the brief but vivid description given by Miranda (1980/1981, p. 9).
 1 2 See (Miranda 1980/1981, p. 9).
 ↑ Miranda (1980/1981, p. 9) remarks precisely that, to perform all those tasks, they could only rely on two old janitors, and that the funds available for the institution were trifling.
 ↑ This highly positive assessment of his work during those years is due to Miranda (1980/1981, p. 9) himself.
 ↑ For the academic year 1944/45, according to De Angelis & Sbordone (1999, p. 29).
 ↑ Miranda (1980/81, p. 9) details briefly but comprehensively these early career steps, while Letta (1981, p. 347) only outlines them. De Angelis & Sbordone (1999, p. 29) state precisely the academic years and the course helds by Cafiero at the Institute.
 1 2 3 See (Miranda 1980/1981, p. 10).
 ↑ De Angelis & Sbordone (1999, p. 29) state that he was lecturer (the exact Italian academic rank was "professore incaricato") of "Matematica generale" (Free English translation:"General mathematics") for the academic year 1952/1953.
 ↑ The "free professorship" (in a literal free English translation) was an academic title similar to the German "Habilitation", no longer in force in Italy since 1970.
 ↑ See Letta (1981, p. 347) and (Miranda 1980/1981, p. 10).
 ↑ See (Miranda 1980/1981, p. 10): Miranda precisely uses the term "carissimo", which in the Italian language means more than dear (caro) and less than dearest (il più caro).
 ↑ See the announce on the Bollettino UMI (1953, p. 471), reporting also the names of other winners and of the judging committee.
 ↑ See Letta (1981, p. 347), (Miranda 1980/1981, p. 10) and the announce on the Bollettino UMI (1953, p. 472), "Nomine di nuovi professori straordinari" section: Letta and Miranda precisely state the month and the year of his appointment.
 1 2 See also the "Teaching activity" section.
 ↑ See (Letta 1981, p. 347), (Marino 2008, p. 2), (Maugeri 1994, p. 179) and (Miranda 1980/1981, p. 10). Letta, Maugeri and Miranda precisely state the month and the year of his arrival: on the other hand, Maugeri and Marino refer also that he substituted Vincenzo Amato (1881–1963), retired during the academic year 1951–1952.
 1 2 According to Maugeri (1994, p. 179) and to Marino (2013, pp. 93–94), who reports a piece of an address by Francesco Guglielmino.
 ↑ See (Letta 1981, p. 347) and (Miranda 1980/1981, p. 11). Letta precisely states that the 1955/1956 academic year was his last one in Catania.
 ↑ As Faedo himself briefly recalls in (Faedo 1986, p. 104).
 ↑ See (Letta 1981, p. 348) and (Miranda 1980/1981, p. 11).
 ↑ According to Letta (1981, p. 348), who refers also that he was awarded a gold medal for the role he played in the construction of a new electronic computer at the university.
 ↑ (English translation) "But it was immediately after the second world war that the process of abstraction of measure and integration theory was completed in a definitive manner. Paul Halmos in the U.S.A. and Renato Caccioppoli, Federico Cafiero (1914–1980) and others in Italy were the main contributors". The Italic type emphasis is due to the Author himself.
 ↑ (English translation) "As Andreotti also Stampacchia could not come immediately to Pisa therefore I was happy to have with me another valiant pupil of Renato Caccioppoli, Federico Cafiero, who was in Pisa for a short time but left a strong trace and formed his valid successor Giorgio Letta."
References
Biographical and general references
 Accademia Pontaniana (2015), Annuario della Accademia Pontaniana 2015 (DLXXIII dalla fondazione) (PDF) (in Italian), Napoli: Nella Sede dell'Accademia, p. 180. The "Yearbook 2015" of the Accademia Pontaniana, published by the Academy itself and describing its past and present hierarchies and its activities. It also gives some notes on its history, the full list of its members and other useful information.
 Bottazzini, Umberto (2010), "La Scuola matematica pisana (1860–1960)" [The Pisan mathematical school (1860–1960)], Annali di Storia delle Università italiane (in Italian), 14.
 de Lucia, Paolo; Sbordone, Carlo (1996), Presentazione (in Italian), p. 9 of the book (Cafiero 1996). The short "Introduction" to Cafiero's selected works by its editors: it includes also a few biographical data.
 De Angelis, P. L.; Sbordone, C., eds. (1999), "Federico Cafiero", Matematici all'Istituto Universitario Navale (1926 – 1976) [Mathematicians at the Istituto Universitario Navale (1926 – 1976)] (in Italian), Napoli: Istituto Universitario Navale/RCE Edizioni, pp. 29–36. The chapter on Cafiero in a book collecting brief biographical sketches and bibliographies of scientific production of the mathematicians who worked at the Parthenope University of Naples, during their stay at the renowned Neapolitan University.
 Faedo, Sandro (1986), "Leonida Tonelli e la scuola matematica pisana", in Montalenti, G.; Amerio, L.; Acquaro, G.; Baiada, E.; et al., Convegno celebrativo del centenario della nascita di Mauro Picone e Leonida Tonelli (6–9 maggio 1985), Atti dei Convegni Lincei (in Italian), 77, Roma: Accademia Nazionale dei Lincei, pp. 89–109. "Leonida Tonelli and the Pisa mathematical school" is a survey of the work of Tonelli in Pisa and his influence on the development of the school, presented at the International congress in occasion of the celebration of the centenary of birth of Mauro Picone and Leonida Tonelli (held in Rome on May 6–9, 1985). The Author was one of his pupils and, after his death, held his chair of mathematical analysis at the University of Pisa, becoming dean of the faculty of sciences and then rector: he exerted a strong positive influence on the development of the university.
 Letta, Giorgio (1981), "Federico Cafiero", Bollettino dell'Unione Matematica Italiana, Sezione A, Serie V, (in Italian), 18 (2): 347–355, MR 0618356, Zbl 0457.01006. Includes a publication list.
 Maugeri, Antonino (1994), "Dal seminario al dipartimento" [From the seminar to the department], Le Matematiche (in Italian), XLIX (I): 175–183, is a short history of the Department of Mathematics of the University of Catania: the Author briefly describes the positive contribution of Federico Cafiero to the research and teaching activity during his stay.
 Marino, Mario (2008), "In Memoria del Prof. Giuseppe Santagati" [In memory of Giuseppe Santagati], Bollettino dell'Accademia Gioenia (in Italian), 41 (369): 1–7, is the obituary of the first doctoral student of Federico Cafiero.
 Marino, Mario (2013), "Breve storia delle cattedre di Analisi matematica dell'Università di Catania nei 150 anni dell'Italia Unitaria" [Short history of the Mathematical analysis chairs at the University of Catania during the 150 years of unified Italy] (PDF), Bollettino dell'Accademia Gioenia (in Italian), 46 (376): 91–105, is the "Last lesson" of Mario Marino. It is a brief historical paper on the chairs of mathematical analysis at the University of Catania: one of these chairs was held by Federico Cafiero for two years, and the paper gives several brief remarks on his work.
 Miranda, Carlo (1978), "Breve storia e prospettive future dell'istituto di matematica della facolta di scienze dell'universita di Napoli" [Short history and future perspectives of the mathematical institute of the faculty of sciences of the University of Neaples], Rendiconto dell'Accademia delle Scienze Fisiche e Matematiche, Napoli, Serie IV, (in Italian), 44 (1977): 7–44, MR 0497758, Zbl 0384.01020.
 Miranda, Carlo (1980–1981), "Federico Cafiero", Rendiconto dell'Accademia delle Scienze Fisiche e Matematiche, Napoli, Serie IV, (in Italian), 47: 9–16, MR 0697874.
 Palladino, Franco; Palladino, Nicla (2009), "I modelli matematici costruiti per l'insegnamento delle matematiche superiori pure e applicate" [Mathematical models built for the teaching of pure and applied higher mathematics] (PDF), Ratio Mathematica (in Italian), 19: 31–88, is a work on the didactics of mathematics, remarking the usefulness of physical models of e.g. curves and surfaces of various kind. It includes a brief remark on the "Bourbakist" style of teaching of Federico Cafiero by the first author.
 Presidenza della Repubblica Italiana (June 9, 1976), Medaglia d'oro ai benemeriti della scuola della cultura e dell'arte: Federico Cafiero, retrieved February 11, 2013.
 Roghi, G. (December 2005), "Materiale per una storia dell'Istituto Nazionale di Alta Matematica dal 1939 to 2003" [Materials toward a history of the Istituto Nazionale di Alta Matematica from 1939 to 2003], Bollettino della Unione Matematica Italiana, Sezione A, La Matematica nella Società e nella Cultura, Serie VIII (in Italian), 8–A (3, parte 2): x+301, MR 2225078, Zbl 1089.01500. This is a monographic fascicle published on the "Bollettino della Unione Matematica Italiana", describing the history of the "Istituto Nazionale di Alta Matematica Francesco Severi" from its foundation in 1939 to 2003. It was written by Gino Roghi and includes a presentation by Salvatore Coen and a preface by Corrado De Concini. It is almost exclusively based on sources from the institute archives: the wealth and variety of materials included, jointly with its appendices and indexes, make this monograph a useful reference not only for the history of the institute itself, but also for the history of many mathematicians who taught or followed the institute courses or simply worked there.
 Società Nazionale di Scienze Lettere e Arti in Napoli (2014), Annuario della Società Nazionale di Scienze Lettere e Arti in Napoli – 2014 (PDF) (in Italian), Napoli: Società Nazionale di Scienze Lettere e Arti in Napoli, p. 82. The "Yearbook 2014" of the Società Nazionale di Scienze Lettere e Arti in Napoli, published by the society itself and describing its past and present hierarchies, and its activities. It also reports some notes on its history, the full list of its members and other useful information.
 UMI (1954), "Notizie" [Notices], Bollettino dell'Unione Matematica Italiana, Serie III (in Italian), 8 (4): 468–487.
 UMI (1956), "Notizie" [Notices], Bollettino dell'Unione Matematica Italiana, Serie III (in Italian), 11 (4): 629–662.
 UMI (1959), "Notizie" [Notices], Bollettino dell'Unione Matematica Italiana, Serie III (in Italian), 14 (4): 583–619.
References describing his scientific contributions
 de Lucia, Paolo (1988), "Analisi reale e teoria della misura a Napoli: R. Caccioppoli, C. Miranda e F. Cafiero", in Società Nazionale di Scienze, Lettere ed Arti in Napoli, Seduta inaugurale dell'anno accademico 1988 (in Italian), Napoli: Francesco Giannini e Figli, pp. 23–33. "Real analysis and measure theory in Naples: R. Caccioppoli, C. Miranda and F. Cafiero" (English translation of the title) is the opening address of the 1988 academic year of the Società Nazionale di Scienze, Lettere ed Arti in Napoli: it describes the contributions of Caccioppoli, Miranda and Cafiero to real analysis and measure theory during their stay in Naples.
 de Lucia, Paolo (2004) [1999], "Teoria della Misura a Napoli: Renato Caccioppoli", in Alvino, A.; Carbone, L.; Sbordone, C.; Trombetti, G., In ricordo di Renato Caccioppoli [In memoriam Renato Caccioppoli] (in Italian) (2nd printing ed.), Napoli: Giannini, p. 124, MR 1306300, Zbl 0793.01019 (reviews of the symposium papers, see below): a collection of papers detailing his personality and his research, including the introduction to his "Opere scelte" (Selected works), a list of contributions from the "International Symposium Renato Caccioppoli" held in Napoli on September 20–22, 1989, a conference held by Caccioppoli himself and related letters by Carlo Miranda, Giovanni Prodi and Francesco Severi. This paper, "Measure theory in Naples: Renato Caccioppoli", appeared in the proceedings of the symposium, details Cacioppoli's and Cafiero's contributions to the development of Measure Theory.
 Fichera, Gaetano (1993), "Il calcolo infinitesimale alle soglie del Duemila" [Infinitesimal calculus at the threshold of the Millennium], Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Supplemento, Serie IX, 4 (1): 69–86, is a survey paper by Gaetano Fichera, describing the development of infinitesimal calculus during the twentieth century and trying to trace possible scenarios for its future evolution.
 Letta, Giorgio (2013), Argomenti scelti di Teoria della Misura [Selected topics in Measure Theory], Quaderni dell'Unione Matematica Italiana (in Italian), 54, Bologna: Unione Matematica Italiana, pp. XI+183, ISBN 8837118805, Zbl 1326.28001, is, according to its Author, an exposition of classical topics in Measure Theory that, despite their conceptual relevance and potential applicability, are rarely taught in current (2012) Italian university courses.
 Piccinini, Livio C.; Stampacchia, Guido; Vidossich, Giovanni (1978), Equazioni differenziali ordinarie in R^{n} (problemi e metodi), Serie di matematica e Fisica "T" (in Italian), 5, Napoli: Liguori Editore, p. 452, ISBN 9788820707286, translated in English as Piccinini, Livio C.; Stampacchia, Guido; Vidossich, Giovanni (1984) [1978], Ordinary differential equations in R^{n}. Problems and methods, Applied Mathematical Sciences, 39, translated by LoBello, A., New York: SpringerVerlag, pp. xii+385, doi:10.1007/9781461251880, ISBN 0387907238, MR 0740539, Zbl 0535.34001.
External links
 Guerraggio, Angelo; Nastasi, Pietro, "Federico Cafiero (1914–1980)", Mathematica Italiana (in Italian), retrieved March 17, 2013.
 O'Connor, John J.; Robertson, Edmund F., "Federico Cafiero", MacTutor History of Mathematics archive, University of St Andrews.