# Gustave Alfred Arthur Choquet

### Born: 1 March 1915 in Solesmes, near Valenciennes, Nord, France

Died: 14 November 2006 in Lyon, France

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**Gustave Choquet**'s father was also named Gustave Choquet and his mother was Marie Fosse. Gustave Choquet Sr was an accountant but had been called up for military duty in July 1914 as the hostilities of World War I began. When Gustave, the subject of this biography, was born, his father was at the front. German troops had invaded Belgium in August 1914 and quickly entered France close to where Marie Choquet together with Gustave's older sister were living. The family had to endure German troops occupying the Valenciennes region when Gustave was born. Eventually the family were evacuated but had to suffer a long journey, first to Switzerland, then back to Paris, before settling in the Vendée region near the west coast of France. This was a peaceful place, well away from the fighting, and Gustave's earliest memories were of happy times in a tiny village in this region. Gustave Choquet Sr survived the war and a third child, a young sister for Gustave, was born. The family returned to the Valenciennes region when the war ended in 1918 living briefly in the town of Valenciennes itself before they moved to the village of Saultain, 5 km to the southeast of the town. There Gustave Sr was employed as an accountant to a textile firm [8]:-

[Gustave attended primary school in Saultain and we know much about his mathematical education from this time on because of his description given in [2]. There he was taught by an exceptional teacher, M Flamant and Choquet wrote that he learnt more from M Flamant than he did at the lycée or at university. M Flamant was particularly keen to demonstrate things visually and he demonstrated many scientific experiments to the young children. However, he carried this over to mathematics and illustrated arithmetical calculations by giving geometrical examples. This made a tremendous impression on the young Choquet and M Flamant's approach to learning and teaching would influence his whole life. There were no secondary schools in the village of Saultain, so Choquet had to attend the lycée at Valenciennes. Again he was fortunate to have some exceptional teachers, particularly M Mas, the mathematics teacher. As with M Flamant, M Mas's [8]:-Gustave's]father had a sense of duty and seriousness about work which made a lasting impression on his son, but he also took great joy in playing the clarinet in the local band. Encouraged by his father, the young Gustave learned to play the flute, and was able to join in concerts and festivals in the local villages. Another occupation shared with his father was the cultivation of the family garden, from which he derived a lifelong love of gardening. His mother contributed great human warmth to the life of the family. She had a taste for music, flowers, and poetry, and herself wrote short poems.

Writing of his mathematical studies at the lycée at Valenciennes, Choquet writes [2]:-... mathematical style he found very congenial and who again encouraged the habit of looking for geometrical approaches to mathematical problems. Choquet derived particular satisfaction at school from solving difficult problems in geometry. Another source of great fascination was an elementary calculus book lent by a classmate, which he studied avidly. Looking back, he thought it could well have been at this time that he decided to make mathematics, if possible, a lifelong pursuit.

While still studying in Valenciennes, Choquet sat the 'concours général', the nation-wide mathematics competition, in 1933 and was ranked in first place beating Roger Apéry who was ranked second. He then went to Paris where he studied at the Lycée Saint-Louis, preparing for his university career. He studied Émile Borel'sA friend had secretly lent me a book 'Analyse de l'École Universelle'Ⓣwhere the derivative was introduced and applied to the study of planar curves. That tempted me to make many drawings and find their tangents, cusps, curvature, which the given formulas allow me to determine; but the habit that I had of starting to read books from the end and zigzag through them, made me miss the definition of symbols y' and y". So I began to reconstruct, thanks to the examples treated, the formalism of the calculus; in that way I could quickly formally calculate the derivatives of polynomials and rational functions. It was a good start in algebra; but unfortunately I quickly understood what a continuous function was, and the charms of analysis were decidedly too large and I definitely swung to that side.

*Leçons sur la théorie des fonctions*Ⓣ which fascinated him. He also studied the works of Gaston Darboux, which he read a little at a time. Later he also read Tannery's works on analysis, finding great pleasure from the Riemann integral, the theorem of Jordan and functions of bounded variation. He took the entrance examinations for the École Normale Supérieure and began his studies there in the autumn of 1934.

At the École Normale Supérieure, Choquet took Georges Valiron's course on analytic functions, and courses by Georges Darmois which he found inspirational. He attended courses on differential geometry given by René Garnier (1887-1984) but it was two books that he discovered in the library which influenced his future research career more than any of these courses. These two books were René Baire's

*Leçons sur les fonctions discontinues*Ⓣ and Georg Cantor's

*Beiträge zur Begründung der transfiniten Mengenlehre*Ⓣ which he read in a French translation. He graduated with the agrégation in 1937 and, following advice from Georges Darmois, spent the summer reading Ernest Hobson's

*Theory of functions of a real variable*and Constantin Carathéodory's

*Vorlesungen über reelle Funktionen*Ⓣ. Here he gained his first introduction to measure theory and to basic general topology. Again following Georges Darmois's advice, in October 1937 he approached Arnaud Denjoy who became his advisor. Certainly Choquet and Denjoy discussed their research but Choquet wanted to follow his own ideas and never asked Denjoy to suggest a research topic. Choquet's research went well and in 1938 he published his first three papers:

*Étude des homéomorphies planes*Ⓣ;

*Étude de certains réseaux de routes*Ⓣ; and

*Prolongement d'homéomorphies*Ⓣ.

Choquet obtained a Jane Eliza Proctor fellowship to finance a year at Princeton. The Jane Eliza Proctor fellowship was a Princeton award to students at the École Normale Supérieure who were of:-

The Jane Eliza Proctor fellowship was for two years but, after spending 1938-39 at Princeton, Choquet had to return to France due to the outbreak of World War II [8]:-... high character, excellent education and exceptional scholarly promise.

Although Choquet had published 28 papers by 1946, he had never submitted a thesis for a doctorate. However, when the opportunity of spending two years at the French Institute in Krakov, Poland, came up with the condition that applicants were required to have a doctorate, he set about rectifying this omission. In three months he had written... he was called up and sent, in the company of a hundred or so former pupils of the grandes écoles, to a camp at Biscarosse for anti-aircraft instruction. His critical attitude to the proposed means of defence earned him an early posting to a horse artillery unit, the71940^{th}R.A.D., and inhe took part in the Battle of France, first in Lorraine, then from the banks of the Aisne to the banks of the Creuse. In August1940he was demobilized at Limoges and shortly afterwards returned to Paris with his fiancée. They were married in January1941. During the war the couple lived frugally in a little two-room apartment in Paris, where their two sons were born. In the years1941-46his research was supported by a C.N.R.S. stipend. The amount was modest, but he was allowed complete freedom to pursue his research as he saw fit, and this proved to be an extremely productive time for him.

*Application des propriétés descriptives de la fonction contingent a la théorie des fonctions de variable réelle et a la géométrie différentielle des variétés cartésiennes.*Ⓣ This work studies:-

Now with a doctorate, his application for the position in Krakov was successful and he went to Poland with his family in 1946. He spent a year during which he made important contacts with Polish mathematicians, in particular Sierpiński, Kuratowski, Steinhaus, and Nikodym on visits to Warsaw and Wrocław. On a visit to Lvov, he added two problems to the Scottish Book. He returned to France in the autumn of 1947 to take up a position as Maître de conférences, an assistant lecturership, at Grenoble. His two sons Bernard and Christian had been born in Paris, and now his third child, his daughter Claire, was born at Grenoble. In 1949 the family moved to Paris when Choquet was appointed as Maître de conférences at the University of Paris. In 1950 he was promoted to professor at the University of Paris. He was, in addition, a Professor at the École Polytechnique from 1960 to 1969. He retired from his Paris chair in 1984.... the differentiability properties of subsets of Euclidean spaces, and is a pioneering contribution to non-smooth analysis which reveals profound relations between certain differentiable and topological structures.

As to his remarkable mathematics contributions, we first quote David A Edwards [9]:-

In 1988, Choquet was elected to Honorary Membership of the London Mathematical Society. Chris Zeeman, President of the Society, read the following citation [6]:-He published more than160mathematical articles and11books. He made important contributions to a variety of fields: topology, measure theory, descriptive set theory, potential theory, and functional analysis. Two themes deserve particular mention. His monumental1953/4paper 'Theory of capacities' contains among many riches his celebrated 'capacitability theorem'. A second great theme is his theory of integral representations in compact convex sets and weakly complete cones, now usually known as 'Choquet theory', which launched a huge development. The work on capacities and that on integral representations have both found many applications in analysis and probability.

The importance of Choquet's work on capacities can be seen from the fact that in 2008 theProfessor Gustave Choquet, who was born in1915, has made major contributions to a wide range of topics in analysis, potential theory, functional analysis, measure theory and infinite dimensional convexity. His work on integral representations over convex sets was so seminal that this area is now known as 'Choquet theory'. Choquet's work on capacities is also particularly striking. The widespread influence of his research has been enhanced by the warmth of his personality. In recognition of the importance of his work he was elected to the Academie des Sciences in1976. He has been Professor of Mathematics at the Universites de Paris VI and XI, and at the École Polytechnique, and is a Chevalier de la Legion d'Honneur.

*International Journal of Approximate Reasoning*produced a 'Special Issue on Choquet integration in honour of Gustave Choquet'. Let us quote from the Foreword:-

InWhereas Lebesgue's classical integration theory is based on s-additive measures, there have been made, during the second half of the201953^{th}century and often independently, many attempts to depart from additivity of the measure, guided by requirements of different applications. One of the earliest and also deepest and most comprehensive of these approaches had been worked out by Gustave Choquet in his/54paper 'Theory of capacities'. He died November14last year in Lyon at the age of91, so this is a good time and place to appreciate those parts of his scientific work related to this issue and to see why, nowadays and with full justification, the integral w.r.t. a monotone non-additive measure is called Choquet integral. In fact, Choquet's famous paper 'Theory of capacities', comprising165pages, is a monograph rather than an ordinary article. It contains the essentials of non-additive measure theory, especially the theory of infinity-alternating set functions and their dual, totally monotone ones, which later have been called belief functions as well. In this context he investigated, in its dual version, the Möbius transform of a non-additive measure under topological assumptions.

*La naissance de la théorie des capacités: réflexion sur une expérience personnelle*Ⓣ (1986), Choquet gives an interesting historical account of the development of the theory of capacities.

Choquet produced a number of important books. His three-volume work

*Cours d'analyse*(1964) was based on his courses at the University of Paris. The three volumes are: 1. Algebra; 2. Topology; and 3. Integration and differential calculus. Another three-volume work

*Lectures on analysis*(1969) was based on Choquet's course in analysis given at Princeton during the autumn term in 1967. The three volumes are: 1. Integration and topological vector spaces; 2. Representation theory; and 3. Infinite dimensional measures and problem solutions. He writes in the Preface to the whole work:-

Another book, again based on a course given by Choquet, was also published in 1969, namelyModern analysis uses a great variety of basic tools, many of which are common to analysts in several branches. I have tried, during a one semester graduate course at Princeton, to present some of those tools that I found useful in potential theory, probability theory and harmonic analysis; they include parts of functional analysis, integration theory and general topology, but I have avoided as much as possible, parts of these theories which, although elegant and interesting in themselves, do not have a large range of applications.

*Outils topologiques et métriques de l'analyse mathématique*.

We noted above Choquet's marriage and his three children. This marriage was dissolved and, on 16 May 1961, he married Yvonne Bruhat. She took the name Yvonne Choquet-Bruhat and has a biography in this archive. Yvonne and Gustave Choquet had a son Daniel and a daughter Geneviève.

We have already mentioned the fact that Choquet was made an honorary member of the London Mathematical Society. However, he received many other honours such as four prizes from the Paris Academy of Sciences, namely the Houllevigue Prize (1945), the Dickson Prize (1951), the Carrière Prize (1956), and the Grand Prix des sciences mathématiques (1968). He was elected a member of the Paris Academy of Sciences in 1976. In 1966 he was made Chevalier of the Légion d'Honneur, later becoming Officier of the Légion d'Honneur. He was also elected to the Bavarian Academy, and received an honorary degree from the Charles University of Prague in 2002.

As to his hobbies, we mention his love of gardening, mountain sports, walking and swimming. We end our biography by quoting David Edwards [8]:-

He will be remembered not only for his fundamental contributions to mathematics, but also as a truly inspiring teacher. His outstanding talents were allied to great kindness and deep humanity, and he won the respect and the warm affection of his many pupils and of scientific colleagues worldwide.

**Article by:** *J J O'Connor* and *E F Robertson*

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**Mathematicians born in the same country**

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