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Über dieses Buch

This textbook provides a detailed treatment of abstract integration theory, construction of the Lebesgue measure via the Riesz-Markov Theorem and also via the Carathéodory Theorem. It also includes some elementary properties of Hausdorff measures as well as the basic properties of spaces of integrable functions and standard theorems on integrals depending on a parameter. Integration on a product space, change of variables formulas as well as the construction and study of classical Cantor sets are treated in detail. Classical convolution inequalities, such as Young's inequality and Hardy-Littlewood-Sobolev inequality are proven. The Radon-Nikodym theorem, notions of harmonic analysis, classical inequalities and interpolation theorems, including Marcinkiewicz's theorem, the definition of Lebesgue points and Lebesgue differentiation theorem are further topics included. A detailed appendix provides the reader with various elements of elementary mathematics, such as a discussion around the calculation of antiderivatives or the Gamma function. The appendix also provides more advanced material such as some basic properties of cardinals and ordinals which are useful in the study of measurability.​

Inhaltsverzeichnis

Frontmatter

Chapter 1. General Theory of Integration

Without Abstract
Nicolas Lerner

Chapter 2. Actual Construction of Measure Spaces

Without Abstract
Nicolas Lerner

Chapter 3. Spaces of Integrable Functions

Without Abstract
Nicolas Lerner

Chapter 4. Integration on a Product Space

Without Abstract
Nicolas Lerner

Chapter 5. Diffeomorphisms of Open Subsets of ℝ n and Integration

Without Abstract
Nicolas Lerner

Chapter 6. Convolution

Without Abstract
Nicolas Lerner

Chapter 7. Complex Measures

Without Abstract
Nicolas Lerner

Chapter 8. Basic Harmonic Analysis on ℝ n

Without Abstract
Nicolas Lerner

Chapter 9. Classical Inequalities

Without Abstract
Nicolas Lerner

Chapter 10. Appendix

Without Abstract
Nicolas Lerner

Backmatter

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