Kirjahaku

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob. Titles in planning include Flavia Smarazzo and Alberto Tesei, Measure Theory: Radon Measures, Young Measures, and Applications to Parabolic Problems (2019) Elena Cordero and Luigi Rodino, Time-Frequency Analysis of Operators (2019) Mark M. Meerschaert, Alla Sikorskii, and Mohsen Zayernouri, Stochastic and Computational Models for Fractional Calculus, second edition (2020) Mariusz Lemanczyk, Ergodic Theory: Spectral Theory, Joinings, and Their Applications (2020) Marco Abate, Holomorphic Dynamics on Hyperbolic Complex Manifolds (2021) Miroslava Antic, Joeri Van der Veken, and Luc Vrancken, Differential Geometry of Submanifolds: Submanifolds of Almost Complex Spaces and Almost Product Spaces (2021) Kai Liu, Ilpo Laine, and Lianzhong Yang, Complex Differential-Difference Equations (2021) Rajendra Vasant Gurjar, Kayo Masuda, and Masayoshi Miyanishi, Affine Space Fibrations (2022)

This book gives a straightforward introduction to the field as it is nowadays required in many branches of analysis and especially in probability theory. The first three chapters (Measure Theory, Integration Theory, Product Measures) basically follow the clear and approved exposition given in the author's earlier book on "Probability Theory and Measure Theory". Special emphasis is laid on a complete discussion of the transformation of measures and integration with respect to the product measure, convergence theorems, parameter depending integrals, as well as the Radon-Nikodym theorem. The final chapter, essentially new and written in a clear and concise style, deals with the theory of Radon measures on Polish or locally compact spaces. With the main results being Luzin's theorem, the Riesz representation theorem, the Portmanteau theorem, and a characterization of locally compact spaces which are Polish, this chapter is a true invitation to study topological measure theory. The text addresses graduate students, who wish to learn the fundamentals in measure and integration theory as needed in modern analysis and probability theory. It will also be an important source for anyone teaching such a course.

Heinz Bauer (1928-2002) was one of the prominent figures in Convex Analysis and Potential Theory in the second half of the 20th century. The Bauer minimum principle and Bauer's work on Silov's boundary and the Dirichlet problem are milestones in convex analysis. Axiomatic potential theory owes him what is known by now as Bauer harmonic spaces. These Selecta collect more than twenty of Bauer's research papers including his seminal papers in Convex Analysis and Potential Theory. Above his research contributions Bauer is best known for his art of writing survey articles. Five of his surveys on different topics are reprinted in this volume. Among them is the well-known article Approximation and Abstract Boundary, for which he was awarded with the Chauvenet Price by the American Mathematical Association in 1980.

_ .... _---------- ------------ Wahrend der letzten zehn Jahre konnte: man eine Neubelebung des Interesses fur die Potentialtheorie beobachten. Zwei Ursachen lassen dies verstandlich erscheinen: Einmal die innere Weiterentwicklung der Potentialtheorie. welche nach der Erfassung moeglichst umfangreicher Klassen von Differentialgleichungen und Kernen drangt, zum anderen die Entwicklung der Theorie der Markoffschen Prozesse und der vor allem durch die bahnbrechende Arbeit von G.A.HUNT erwirkte Bruckenschlag hinuber zur Potentialtheorie. Die genannte innere Entwicklung der Potentialtheorie hat, aufbauend auf Ideen von TAUTZ I} 9], I} 0], DOOB [!9] und BRELOT, zu einer Axiomatisierung der Theorie der harmonischen Funktionen ge- fuhrt mit dem Ziel eines gleichzeitigen Erfassens bereits vorliegen- der Resultate uber die Potentialtheorie RieTrlannscher Flachen und Greenscher Raume und einer Ausdehnung der Potentialtheorie der Laplace-Gleichung auf bislang unerforschte Klassen elliptischer Differentialgleichungen. A: m bekanntesten und a: m weitesten vollendet ist in dieser Richtung die in OS] dargestellte Theorie von BRELOT. Wichtige Erganzungen verdankt man der These 1}1] von MadaTrle, HERVE - Wahrend die Brelotsche Theorie ausschliesslich elliptische Gleichungen betrifft, bemuhten sich DOOB o]. KAMKE {1 und Verf. um die Einbeziehung auch parabolischer partieller Diffe- rentialgleichungen zweiter Ordnung.