A Course on Integration Theory - Grand Format

This textbook provides a detailed treatment of abstract integration theory, construction of the Lebesgue measure via the Riesz-MarkovTheorem 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 attd Hardy-Littlewood-Sobolev inequality, are proven. Further topics include the Radott-Nikodym theorem, notions of hamtonic analysis, classical inequalities and interpo lation theorems including Marcinkiewiez's theorem, and the definition of Lebesgue points and the Lebesgue differentiation theorem. Each chapter ends with a large number of exercises and detailed solutions.
A comprehensive appendix provides the reader with various elements of elementary mathematics, such as a discussion around the calculation of antiderivatives or the Gamma function. It also provides more advanced Material such as some basic properties of cardinals and ordinals which are useful for the study of measurability.