Khóa luận độ đo Hausdorff

Felix Hausdorff`s father was Louis Hausdorff, who was a merchant dealing in textiles, and his mother was Hedwig Tietz; both were Jewish. Felix was born into a wealthy family and this had quite an influence on his life and career since he never had the problem of having to work to support himself financially. Felix was stll a young boy when the family moved from Breslau to Leipzig, and it was in Leipzig that he grew up. At school he had wide interests and, in addition to mathematics, he was attracted to literature and music. In fact he wanted to pursue a career in music as a composer but his parents put pressure on him to give up the idea of becoming a composer. They achieved this, but only after quite an effort for Felix had his heart set on the idea, and after this he turned towards mathematics as the subject to study at university.
Hausdorff studied at Leipzig University under Heinrich Bruns and Adolph Mayer, graduating in 1891 with a doctorate in applications of mathematics to astronomy. His thesis was titled Zur Theorie der astronomischen Strahlenbrechung and studied refraction and extinction of light in the atmosphere. He published four papers on astronomy and optics over the next few years and he submitted his habilitation thesis to Leipzig in 1895, also based on his research into astronomy and optics. His methods were based on those of Bruns who had developed his own method of determining refraction and extinction, based on an idea of Bessel.
However Hausdorff`s main interests were in literature and philosophy and his circle of friends consisted almost entirely of writers and artists, such as the composer Max Reger, rather than scientists. He also seemed keen to make a name for himself in the world of literature, more so than in the world of mathematics, and he published his literary work under the pseudonym of Paul Mongré. In 1897 he published his first literary work Sant` Ilario: Thoughts from Zarathustra`s Country which was a work of 378 pages. He published a philosophy book Das Chaos in kosmischer Auslese (1898) which is a critique of metaphysics contrasting the empirical with the transcendental world that he rejected. His next major literary work was a book of poem Ekstases (1900) which deals with nature, life, death and erotic passion, and in addition he wrote many articles on philosophy and literature. As Segal writes in [6]:-
As the child of a wealthy family, he did not have to worry about making a career as a mathematician; for him, mathematics, both as research and as a subject to teach, was more an avocation than anything else.
Hausdorff married Charlotte Sara Goldschmidt in Leipzig in 1899. Charlotte and her sister Edith were from Jewish parents but had converted to Lutheranism. Although still a Privatdozent, Hausdorff was well off, so marriage at this stage in his career presented no financial difficulties. In 1902 he was promoted to an extraordinary professorship of mathematics at Leipzig and turned down the offer of a similar appointment at Göttingen. This clearly indicates that at this time Hausdorff was keener to remain in his literary and artistic circle in Leipzig than he was to progress his career in mathematics. He continued his literary interests and in 1904 published a farce Der Arzt seiner Ehre. In many ways this marked the end of his literary interests but this farce was performed in 1912 and was very successful.
After 1904 Hausdorff began working in the area for which he is famous, namely topology and set theory. He introduced the concept of a partially ordered set and from 1901 to 1909 he proved a series of results on ordered sets. In 1907 he introduced special types of ordinals in an attempt to prove Cantor`s continuum hypothesis. He also posed a generalisation of the continuum hypothesis by asking if 2 to the power ℵa was equal to ℵa+1. Hausdorff proved further results on the cardinality of Borel sets in 1916.
Hausdorff taught at Leipzig until 1910 when he went to Bonn. It was Study who in many ways motivated Hausdorff to become more involved in both mathematical research and also in developing his career in mathematics. Partly the lack of mathematical drive in his early career had been due to his extreme modesty, so his friendship with Study was an important factor in turning him towards important problems and his subsequent rise to fame. Having encouraged Hausdorff to move to Bonn, Study encouraged him to move again in 1913, this time to become an ordinary professorship at Greifswalf. A year later, in 1914, Hausdorff published his famous text Grundzüge der Mengenlehre which builds on work by Fréchet and others to created a theory of topological and metric spaces. Earlier results on topology fitted naturally into the framework set up by Hausdorff as Katetov explains in [1]:-
[Hausdorff`s] broad approach, his aesthetic feeling, and his sense of balance may have played a substantial part. He succeeded in creating a theory of topological and metric spaces into which the previous results fitted well, and he enriched it with many new notions and theorems. From the modern point of view, the Grundzüge contained, in addition to other special topics, the beginnings of the theories of topological and metric spaces, which are now included in all textbooks on the subject.
The Grundzüge was republished in revised form in 1927 and