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.