昂利·勒贝格Henri Lebesgue

昂利·勒贝格的父亲是一名印刷工。勒贝格在博韦学院开始学业，然后前往巴黎，先是在圣路易中学学习，然后在路易大帝中学学习。

勒贝格于1894年进入巴黎高等师范学校，并于1897年获得数学教学文凭。在接下来的两年里，他在该校图书馆学习，阅读了勒内-路易·贝尔关于不连续函数的论文，并意识到在这一领域可以取得更多成就。后来，勒内-路易·贝尔和勒贝格之间会有相当大的竞争，我们在下面提到。他被任命为南锡中央中学的教授，从1899年到1902年在那里任教。在他人工作的基础上，包括埃米尔·博雷尔和卡米耶·若尔当的工作，勒贝格于1901年提出了测度论，并在其著名论文Sur une généralisation de l'intégrale définie Ⓣ（论定积分的一种推广）中，该论文于1901年4月29日发表在Comptes Rendus上，他给出了勒贝格积分的定义，通过将曲线下面积的概念扩展到包括许多不连续函数，推广了波恩哈德·黎曼积分的概念。这种对波恩哈德·黎曼积分的推广彻底改变了积分学。直到19世纪末，数学分析仅限于连续函数，主要基于波恩哈德·黎曼的积分方法。

他的贡献是现代分析的成就之一，极大地扩展了傅里叶分析的范围。这项杰出的工作出现在勒贝格的博士论文Intégrale, longueur, aire Ⓣ（积分、长度、面积）中，于1902年提交给巴黎理学院，这篇130页的著作于同年在米兰的Annali di Matematica上发表。获得博士学位后，勒贝格于1902年获得他的第一个大学职位，成为雷恩理学院的数学讲师。这符合法国年轻学者的标准传统，即先在各省任职，然后后来获得认可，被任命到巴黎的一个更初级的职位。1903年12月3日，他与Louise-Marguerite Vallet结婚，他们有两个孩子。然而，这段婚姻只持续到1916年，他们离婚了。

勒贝格在职业生涯早期获得的一项荣誉是受邀在法兰西学院讲授Peccot课程。他在1903年这样做了，然后两年后的1905年再次受邀讲授Peccot课程。勒贝格在1904年首次与勒内-路易·贝尔发生争执，当时勒内-路易·贝尔在法兰西学院讲授Peccot课程，争论谁最有资格教授这样的课程。他们的竞争在他们后来的生活中变成了更严重的争论。勒贝格写了两本专著Leçons sur l'intégration et la recherche des fonctions primitives Ⓣ（积分讲义与原函数研究）（1904年）和Leçons sur les séries trigonométriques Ⓣ（三角级数讲义）（1906年），这些专著源于这两门课程，并有助于使他的重要思想更广泛地为人所知。然而，他的工作受到经典分析学家的敌视，尤其是在法国。1906年，他被任命到普瓦捷理学院，次年，他被任命为那里的力学教授。

让我们尝试指出勒贝格积分如何使许多与积分相关的问题得以解决。约瑟夫·傅里叶曾假设对于有界函数，表示该函数的无穷级数可以逐项积分。由此，他能够证明，如果一个函数可以用三角级数表示，那么这个级数必然是它的约瑟夫·傅里叶级数。这里有一个问题，即一个不是波恩哈德·黎曼可积的函数可以表示为一致有界的波恩哈德·黎曼可积函数级数。这表明约瑟夫·傅里叶对有界函数的假设不成立。

1905年，勒贝格深入讨论了鲁道夫·利普希茨和卡米耶·若尔当为确保函数$f (x)$是其约瑟夫·傅里叶级数之和而使用的各种条件。勒贝格能够证明的是，对一致有界的勒贝格可积函数级数逐项积分总是有效的。这现在意味着约瑟夫·傅里叶的证明——如果一个函数可由三角级数表示，那么这个级数必然是它的约瑟夫·傅里叶级数——变得有效，因为它现在可以建立在关于级数逐项积分的正确结果之上。正如Hawkins在[1]中所写：

勒贝格的工作……积分的广义定义只是他对积分理论贡献的起点。使这个新定义重要的是，勒贝格能够从中认识到一种分析工具，能够处理——并在很大程度上克服——与波恩哈德·黎曼的积分理论相关的众多理论困难。事实上，这些困难所提出的问题激发了勒贝格的所有主要结果。

1910年，他被任命为索邦大学数学分析讲师。第一次世界大战期间，他为法国国防工作，此时他与从事类似任务的埃米尔·博雷尔发生了争执。勒贝格在索邦大学担任该职位直到1918年，当时他被提升为几何在分析中的应用教授。1921年，他被任命为法兰西公学院数学教授，一直担任该职位直到1941年去世。他还在1927年至1937年间在巴黎市立物理与工业化学高等学院任教，并在塞夫勒的巴黎高等师范学校任教。

有趣的是，勒贝格并没有在整个职业生涯中专注于他自己开创的领域。这是因为他的工作是一个惊人的推广，然而勒贝格本人却害怕推广。他写道：

若被化简为一般理论，数学将是一种没有内容的美丽形式。它会很快消亡。

尽管后来的发展表明他的担忧毫无根据，但这些发展确实让我们得以理解他自己工作所遵循的路径。

他还在数学的其他领域做出了重大贡献，包括 拓扑学、potential theory、约翰·彼得·古斯塔夫·勒热纳·狄利克雷 问题、变分法、集合论、表面积理论和维数理论。到 1922 年他发表 Notice sur les travaux scientifique de M Henri Lebesgue 时，他已经写了近 90 部著作和论文。这部九十二页的著作还提供了对 勒贝格 论文内容的分析。1922 年之后他仍然活跃，但他的贡献转向了教学问题、历史工作和初等几何。

勒贝格 荣获许多科学院选举为院士。他于 1922 年 5 月 29 日当选为 科学院，还当选为 皇家学会、Royal Academy of Science and Letters of Belgium（1931 年 6 月 6 日）、博洛尼亚科学院、Accademia dei Lincei、Royal Danish Academy of Sciences、Romanian Academy of Sciences 以及克拉科夫科学与文学科学院院士。他还被许多大学授予荣誉博士学位。他还获得了多项奖项，包括 Prix Houllevigue（1912 年）、Prix 让-维克托·彭赛列（1914 年）、Prix Saintour（1917 年）和 Prix Petit d'Ormoy（1919 年）。

Henri Lebesgue's father was a printer. Henri began his studies at the Collège de Beauvais, then he went to Paris where he studied first at the Lycée Saint Louis and then at the Lycée Louis-le-Grand.

Lebesgue entered the École Normale Supérieure in Paris in 1894 and was awarded his teaching diploma in mathematics in 1897. For the next two years he studied in its library where he read Baire's papers on discontinuous functions and realised that much more could be achieved in this area. Later there would be considerable rivalry between Baire and Lebesgue which we refer to below. He was appointed professor at the Lycée Centrale at Nancy where he taught from 1899 to 1902. Building on the work of others, including that of Émile Borel and Camille Jordan, Lebesgue formulated the theory of measure in 1901 and in his famous paper Sur une généralisation de l'intégrale définie Ⓣ, which appeared in the Comptes Rendus on 29 April 1901, he gave the definition of the Lebesgue integral that generalises the notion of the Riemann integral by extending the concept of the area below a curve to include many discontinuous functions. This generalisation of the Riemann integral revolutionised the integral calculus. Up to the end of the 19th century, mathematical analysis was limited to continuous functions, based largely on the Riemann method of integration.

His contribution is one of the achievements of modern analysis which greatly expands the scope of Fourier analysis. This outstanding piece of work appears in Lebesgue's doctoral dissertation, Intégrale, longueur, aire Ⓣ , presented to the Faculty of Science in Paris in 1902, and the 130 page work was published in Milan in the Annali di Matematica in the same year. Having graduated with his doctorate, Lebesgue obtained his first university appointment when in 1902 he became mâitre de conférences in mathematics at the Faculty of Science in Rennes. This was in keeping with the standard French tradition of a young academic first having appointments in the provinces, then later gaining recognition in being appointed to a more junior post in Paris. On 3 December 1903 he married Louise-Marguerite Vallet and they had two children. However the marriage only lasted until 1916 when they were divorced.

One honour which Lebesgue received at an early stage in his career was an invitation to give the Cours Peccot at the Collège de France. He did so in 1903 and then received an invitation to present the Cours Peccot two years later in 1905. Lebesgue first fell out with Baire in 1904, when Baire gave the Cours Peccot at the Collège de France, over who had the most right to teach such a course. Their rivalry turned into a more serious argument later in their lives. Lebesgue wrote two monographs Leçons sur l'intégration et la recherche des fonctions primitives Ⓣ (1904) and Leçons sur les séries trigonométriques Ⓣ (1906) which arose from these two lecture courses and served to make his important ideas more widely known. However, his work received a hostile reception from classical analysts, especially in France. In 1906 he was appointed to the Faculty of Science in Poitiers and in the following year he was named professor of mechanics there.

Let us attempt to indicate the way that the Lebesgue integral enabled many of the problems associated with integration to be solved. Fourier had assumed that for bounded functions term by term integration of an infinite series representing the function was possible. From this he was able to prove that if a function was representable by a trigonometric series then this series is necessarily its Fourier series. There is a problem here, namely that a function which is not Riemann integrable may be represented as a uniformly bounded series of Riemann integrable functions. This shows that Fourier's assumption for bounded functions does not hold.

In 1905 Lebesgue gave a deep discussion of the various conditions Lipschitz and Jordan had used in order to ensure that a function $f (x)$ is the sum of its Fourier series. What Lebesgue was able to show was that term by term integration of a uniformly bounded series of Lebesgue integrable functions was always valid. This now meant that Fourier's proof that if a function was representable by a trigonometric series then this series is necessarily its Fourier series became valid, since it could now be founded on a correct result regarding term by term integration of series. As Hawkins writes in [1]:-

In Lebesgue's work ... the generalised definition of the integral was simply the starting point of his contributions to integration theory. What made the new definition important was that Lebesgue was able to recognise in it an analytic tool capable of dealing with - and to a large extent overcoming - the numerous theoretical difficulties that had arisen in connection with Riemann's theory of integration. In fact, the problems posed by these difficulties motivated all of Lebesgue's major results.

He was appointed mâitre de conférences in mathematical analysis at the Sorbonne in 1910. During the first world war he worked for the defence of France, and at this time he fell out with Borel who was doing a similar task. Lebesgue held his post at the Sorbonne until 1918 when he was promoted to Professor of the Application of Geometry to Analysis. In 1921 he was named as Professor of Mathematics at the Collège de France, a position he held until his death in 1941. He also taught at the École Supérieure de Physique et de Chimie Industrielles de la Ville de Paris between 1927 and 1937 and at the École Normale Supérieure in Sèvres.

It is interesting that Lebesgue did not concentrate throughout his career on the field which he had himself started. This was because his work was a striking generalisation, yet Lebesgue himself was fearful of generalisations. He wrote:-

Reduced to general theories, mathematics would be a beautiful form without content. It would quickly die.

Although future developments showed his fears to be groundless, they do allow us to understand the course his own work followed.

He also made major contributions in other areas of mathematics, including topology, potential theory, the Dirichlet problem, the calculus of variations, set theory, the theory of surface area and dimension theory. By 1922 when he published Notice sur les travaux scientifique de M Henri Lebesgue he had written nearly 90 books and papers. This ninety-two page work also provides an analysis of the contents of Lebesgue's papers. After 1922 he remained active, but his contributions were directed towards pedagogical issues, historical work, and elementary geometry.

Lebesgue was honoured with election to many academies. He was elected to the Academy of Sciences on 29 May 1922, to the Royal Society, the Royal Academy of Science and Letters of Belgium (6 June 1931), the Academy of Bologna, the Accademia dei Lincei, the Royal Danish Academy of Sciences, the Romanian Academy of Sciences, and the Kraków Academy of Science and Letters. He was also awarded honorary doctorates from many universities. He also received a number of prizes including the Prix Houllevigue (1912), the Prix Poncelet (1914), the Prix Saintour (1917) and the Prix Petit d'Ormoy (1919).