Once he had completed his schooling, Lebesgue was required to undertake military service and he enlisted in the army at the end of 1809. He spent a year in the army, employed in the offices of the Administration. The system as it operated in France at this time, called 'remplacement militaire', allowed wealthy citizens to pay a sum of money, essentially the amount required to pay for someone else to serve in their place, to avoid military service. After serving for one year, Lebesgue decided he wanted to return to civilian life and the only way to do that was via the 'remplacement militaire' system. Through sacrifices that he made himself and with the help of one of his parents, he was able to pay the necessary sum and in 1810 returned to civilian life. He studied in Paris and was awarded the degree of Bachelor of Science in 1813.
After graduating, Lebesgue was appointed as a master at the Lycée at Reims. Moving on, over the following years he taught next at the College of Saint-Quentin in 1814 and then at the College of Abbeville in 1817 where he taught for three of four years. He then left school teaching, taking a position as a tutor to an English family. He moved to London where he acted as a tutor to the family for one year. Continuing in his career as a private tutor, Lebesgue next took a position as a tutor to a Russian family and he moved to Russia where he stayed until 1830. While in Russia he had been undertaking research in number theory and he published three papers with the title Extrait d'un Mémoire inédit sur les congruences d'un degré quelconque, et à une seule inconnue . These were published in the first two volumes of the Bulletin du Nord, published in Moscow in 1829. The three papers appeared in the January 1829, March 1829 and May 1829 parts of the Bulletin. He returned to France in 1830.
At this stage in his career Lebesgue was helped by his friend Charles Alexandre who was head of the Royal Bourbon College and later, in 1840, was appointed inspector-general of studies in place of M Poullet-Delise who retired at this time. Alexandre was a friend of Siméon-Denis Poisson who, at this time, was a member of the Royal Council of Public Instruction, of the Directorate General of Mathematical Education. Alexandre asked Poisson if he could arrange a position for Lebesgue, giving him an excellent recommendation. Poisson arranged for Lebesgue to obtain a position as a lecturer at the Royal College of Nantes, then in the following year as professor at the college of Epinal, where he remained for two years.
In 1834, he went to Paris to attend the Sorbonne, with the intention of competing for the agrégation. The agrégation is a competitive examination, essentially the same standard as the German habilitation, which leads to a professorship in higher education. Lebesgue had obtained leave from his position to allow him to follow the two year course. In spite of the entreaties of his friends, he did not proceed with this project, and was sent to Neufchateau; he spent the years 1835 and 1836 there. Lebesgue was very active in research and published many papers. He corresponded with Poisson, who was very impressed by the quality and depth of Lebesgue's work.
Antoine Cournot had obtained a position with the Academy of Sciences in 1833, arranged by Poisson. Again it was due to Poisson's recommendation that Cournot was appointed to a newly created chair in analysis at Lyon in 1834. In 1835 Cournot became professor of mathematics at Grenoble and rector there. A substitute for Cournot at Grenoble was required and Lebesgue was appointed to this position. He had, at the age of 46, at last become a teacher at a university. Of course, he had never obtained a doctorate, but this was quickly achieved when he submitted Thèses de Mécanique et d'Astronomie to the University of Grenoble in 1837. The thesis was in two parts, the first being Formules pour la transformation des fonctions homogènes du second degré à plusieurs inconnues while the second, much shorter part, was Applications.
The Faculty of Sciences at Bordeaux was organized at the end of 1838, and Lebesgue was called upon to occupy the Chair of Pure Mathematics. He remained in that position for twenty years until the end of 1858, when he decided to retire. In 1839 he had been ranked third in line for a place as corresponding member in the Geometry Section of the Academy of Sciences; M Chasles, who was ranked in the first place, was elected on this occasion. A new vacancy occurred in 1847 and Lebesgue was unanimously elected at the meeting which took place on 8 February 1847. Earlier, he had received the decoration of the Légion d'Honneur in 1845.
One of his friends was the number theorist Eugène Prouhet (1817-1867) who had been a student of Charles-François Sturm at the École Polytechnique in Paris. Prouhet assisted Lebesgue in the preparation of his book Exercices d'Analyse numérique. Extraits, Commentaires et Recherches relatifs à l'Analyse indéterminée et à la Théorie des Nombres (Libraire Centrale des Sciences, Paris, 1859). In it he writes:-
The various parts of these Exercises being brought together might, by means of some modifications, form an elementary treatise on the theory of numbers.He hoped that sufficient interest would be shown and subscribers found to fund the publication of a second part but this was not the case. However, prince Alphonse de Polignac (1826-1863), himself a number theorist, supported the publication of Lebesgue's second book Introduction à la Théorie des Nombres (Mallet-Bachelier, Paris, 1862). Lebesgue explains these circumstances in the Preface to the work:-
As early as 1858, when I left my duties as a professor at the Faculty of Sciences at Bordeaux, I had resolved to devote my leisure to this work. I then published Exercices d'Analyse numérique, and I announced the publication by subscription of the book in question. The small number of subscribers did not allow this intention to be carried out, and I had relinquished it with regret, when prince Alphonse de Polignac (1826-1863), who is himself engaged in number theory research, as can be seen from the reports of the sessions of the Academy of Sciences, was kind enough to remove most of the obstacles that prevented me. My gratitude for this service will undoubtedly be shared by all those who, like the prince, think that a clear exposition of the present state of the theory of numbers, especially since it is well-established, cannot fail to serve for the advancement of pure mathematics.
Although Lebesgue had only spent 20 years as professor at Bordeaux, nevertheless by the time he retired in 1858 he had published 80 mathematics articles.
In fact after retiring from Bordeaux, Lebesgue returned to Paris where he spent the years from 1858 to 1861 working on the two books we have just mentioned. It was while he was in Paris that he was able to get the support he needed from prince Alphonse de Polignac. Despite the fact that he was working on his books during these years in Paris, Lebesgue still managed to publish three papers in 1858, seven papers in 1859, and three papers in 1860. In 1861 he returned to Bordeaux, mainly because he found the winters in Paris too severe and wished to return to the warmer climes of Bordeaux. Although he was now retired, nevertheless once back in Bordeaux he again lectured at the University. After he had retired from the Chair of Pure Mathematics at Bordeaux, Jules Hoüel had been appointed to fill the chair which he held from 1859 until his death in 1886. Lebesgue and Hoüel collaborated and Hoüel did much to help Lebesgue in preparing Tables diverses pour la Décomposition des Nombres en leurs Facteurs premiers (Gauthier-Villars, Paris, 1864). Lebesgue writes in the Introduction:-
M Hoüel, Professor at the Faculty of Science of Bordeaux, was kind enough to take charge of the construction of this table, according to my indications. He has also lent me his help in coordinating the materials contained in this Memoir, and for the correction of the proofs.In this work, he gives tables of divisors of numbers from 1 to 115500 constructed by sieve methods.
After two years at Bordeaux, when he was working on his Tables, Lebesgue again went to Paris but after only a short stay he again returned to south west France, living in Bordeaux. He then moved around, keeping to south west France, living in towns such as Agen which is 135 km south east of Bordeaux, Augouleme, which is north east of Bordeaux, Dax, a spar town south of Bordeaux, and Pau, even further south near the Pyrenees and not far from the Spanish border. He spent the last twelve years of his life in these south east towns where the climate was to his liking; wet and mild in the winter and warm in the summer. He continued to undertake research and published 23 papers during these final years of his life. These include questions he posed and solutions to questions of others that he had solved. Two of these papers are posthumous, one of which is included in  (but not in ) in which Lebesgue gives an overview of his own research contributions. He divides his contributions into: Higher Algebra; Infinitesimal Analysis; and Number Theory, this last being by far the most substantial of the three sections.
While in Pau in May 1875 he was struck down with a serious illness with alarming symptoms. His daughter hurried to his side and arranged for him to be brought back to Bordeaux. He died in Bordeaux at nine o'clock in the evening on 10 June.
He is described in  (and ) as follows:-
Very simple manners, of a character full of frankness, and independence, virtuous at every test, never seeking the opportunities of putting himself to the fore, Lebesgue lived a very solitary life, constantly occupied with his favourite studies.
Article by: J J O'Connor and E F Robertson