Editing Felix Hausdorff (section)

===Descriptive set theory, measure theory and analysis===
In 1916, Alexandroff and Hausdorff independently solved<ref>P. Alexandroff: ''Sur la puissance des ensembles mesurables B.'' Comptes rendus Acad. Sci. Paris 162 (1916), S.&nbsp;323–325.</ref> the continuum problem for Borel sets: Every Borel set in a complete separable metric space is either countable or has the cardinality of the continuum. This result generalizes the [[Cantor–Bendixson theorem]] that such a statement holds for the closed sets of <math>\mathbb{R}^n</math>. For linear <math>G_{\delta}</math> sets [[William Henry Young]] had proved the result in 1903,<ref>W. H. Young: ''Zur Lehre der nicht abgeschlossenen Punktmengen''. Berichte über die Verhandlungen der Königl. Sächs. Ges. der Wiss. zu Leipzig, Math.-Phys. Klasse&nbsp;55 (1903), S.&nbsp;287–293.</ref> for <math>G_{\delta\sigma\delta}</math> sets Hausdorff obtained a corresponding result in 1914 in ''Principles''. The theorem of Alexandroff and Hausdorff was a strong impetus for further development of descriptive set theory.<ref>Alexandorff, Hopf 1935, S.&nbsp;20. For details see ''Gesammelte Werke Band&nbsp;II'', S.&nbsp;773–787.</ref>

Among the publications of Hausdorff in his time at Greifswald the work ''Dimension and outer measure'' from 1919 is particularly outstanding. In this work, the concepts were introduced which are now known as [[Hausdorff measure]] and the [[Hausdorff dimension]]. It has remained highly topical and in later years has been one of the most cited mathematical works from the decade of 1910 to 1920.

The concept of Hausdorff dimension is useful for the characterization and comparison of "highly rugged quantities". The concepts of ''Dimension and outer measure'' have experienced applications and further developments in many areas such as in the theory of dynamical systems, geometric measure theory, the theory of self-similar sets and fractals, the theory of stochastic processes, harmonic analysis, potential theory, and number theory.<ref>For the history of the reception of ''Dimension und äußeres Maß'', see the article by Bandt/Haase and Bothe/Schmeling in Brieskorn 1996, S.&nbsp;149–183 and S.&nbsp;229–252 and the commentary of S.&nbsp;D.&nbsp;Chatterji in ''Gesammelten Werken, Band&nbsp;IV'', S.&nbsp;44–54 and the literature given there.</ref>

Significant analytical work of Hausdorff occurred in his second time at Bonn. In ''Summation methods and moment sequences I'' in 1921, he developed a whole class of summation methods for divergent series, which today are called [[Hausdorff method]]s. In [[Godfrey Harold Hardy|Hardy]]'s classic ''Divergent Series'', an entire chapter is devoted to the Hausdorff method. The classical methods of [[Otto Hölder|Hölder]] and [[Ernesto Cesàro|Cesàro]] proved to be special cases of the Hausdorff method. Every Hausdorff method is given by a moment sequence; in this context Hausdorff gave an elegant solution of the moment problem for a finite interval, bypassing the theory of continued fractions. In his paper ''Moment problems for a finite interval'' of 1923 he treated more special moment problems, such as those with certain restrictions for generating density <math>\varphi(x)</math>, for instance <math>\varphi(x) \in L^p[0,1]</math>. Criteria for solvability and decidability of moment problems occupied Hausdorff for many years, as hundreds of pages of handwritten notes in his [[Nachlass]] attest.<ref>''Gesammelte Werke Band&nbsp;IV'', S.&nbsp;105–171, 191–235, 255–267 and 339–373.</ref>

A significant contribution to the emerging field of functional analysis in the 1920s was Hausdorff's extension of the [[Riesz-Fischer theorem]] to <math>L^p</math> spaces in his 1923 work ''An extension of Parseval's theorem on Fourier series''. He proved the inequalities now named after him and [[W.H. Young]]. The Hausdorff–Young inequalities became the starting point of major new developments.<ref>See commentary by S.&nbsp;D.&nbsp;Chatterji in ''Gesammelten Werken Band&nbsp;IV'', S.&nbsp;182–190.</ref>

Hausdorff's book ''Set Theory'' appeared in 1927. This was declared as a second Edition of ''Principles'', but it was actually a completely new book. Since the scale was significantly reduced due to its appearance in Goschen's teaching library, large parts of the theory of ordered sets and measures and integration theory were removed. In its preface, Hausdorff writes, "Perhaps even more than these deletions the reader will regret the most that, to further save space in point set theory, I have abandoned the topological point of view through which the first edition has apparently acquired many friends, and focused on the simpler theory of metric spaces".

In fact, this was an explicit regret of some reviewers of the work. As a kind of compensation Hausdorff showed for the first time the then-current state of descriptive set theory. This fact assured the book almost as intense a reception as ''Principles'', especially in Fundamenta Mathematicae. As a textbook it was very popular. In 1935 there was an expanded edition published, and this was reprinted by Dover in 1944. An English translation appeared in 1957 with reprints in 1962 and 1967.

There was also a Russian edition (1937), although it was only partially a faithful translation, and partly a reworking by Alexandroff and [[Kolmogorov]]. In this translation the topological point of view again moved to the forefront. In 1928 a review of ''Set Theory'' was written by Hans Hahn, who perhaps had the danger of German antisemitism in his mind as he closed his discussion with the following sentence:

{{blockquote|An exemplary depiction in every respect of a difficult and thorny area, a work on par with those which have carried the fame of German science throughout the world, and such that all German mathematicians may be proud of.<ref>{{cite journal | last1 = Hahn | first1 = H. | year = 1928 | title = F. Hausdorff, Mengenlehre | journal = [[Monatshefte für Mathematik und Physik]] | volume = 35 | pages = 56–58 }}</ref>}}