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Description
This textbook provides a thorough exposition of measure and integration theory in a style that is very detailed and thus easy to follow and understand. While breaking down argumentation to simple steps and maintaining a consistent exposition, the book presents the theory on a high level of scientific rigor and generality.
Although the book covers a wide spectrum of topics, it leads to a path in functional analysis and probability theory. Thus, a full chapter on measures with densities, including Lebesgue decomposition and Radon-Nikodym's theorem, is included, as well as a chapter on transformation of measures, which is a crucial topic in probability theory. Chapters on Hilbert spaces and the Fourier transform prepare the way for a direction in functional analysis. The book features also a presentation of the Cauchy-Stieltjes transform, which is an alternative to the Fourier transform and highly used in some branches of mathematics, like mathematical physics and random matrix theory.
The textbook has been tested for more than a decade in classrooms. Its style is particularly aimed at students who prefer a step-by-step approach, making it suitable for self-study as well. Appendices that cover all needed background material on real numbers, countability and metric spaces, are provided.
-algebra and measure.- Dynkin s Lemmaanduniqueness of measures.- Constructionofmeasures.- Measurable functions and mappings.- The Lebesgue integral.- Product measures.- Integral Inequalities and Lp-spaces.- Measurability and integration for complex-valued functions.- Hilbert Spaces.- Densities and Absolute Continuity.- Transformation of measures.- The FourierTransform.- Fundamentals of probability theory.- The Cauchy-Stieltjes Transform.
Steen Thorbjørnsen earned his Ph.D. in mathematics from the University of Southern Denmark in 1998 with a specialization in operator algebras. His subsequent research has focused on classical and free probability theory and the interplay between these areas. In 2005 he joined Aarhus University as an associate professor, where he has since taught a wide range of courses in measure and integration theory, probability theory, functional analysis, and operator algebras.
