This book offers a rigorous, comprehensive, and modern presentation of the most traditional concepts in measure theory and integration. Building on the classical foundations, it introduces the theory with full generality and meticulous attention to detail, following the stylistic tradition first introduced by Nicolas Bourbaki. The book is designed for graduate students and young researchers seeking a thorough exposition of the theory in an abstract setting, complete proofs, and the strategies underlying them, fostering good mathematical habits in logical reasoning and clarity of deduction. Beyond standard treatments, the book features several distinctive elements: Some classical results, such as Radon-Nikodým theorem, and Lebesgue and Hahn decompositions, have been presented with original proofs, aimed to clarifying the logic behind the results; some topics that are often overlooked, such as kernels, uniform integrability, the Vitali-Hahn-Saks and Dunford-Pettis theorems are developed in full in dedicated chapters, and a full account of the disintegration of measures is developed. The book also pays special attention to modern applications, including the construction of product measures for an arbitrary family of measures, by exploiting the properties of kernels, a full account of Daniell’s and Carathéodory’s methods for constructing and extending measures, and a thorough coverage of the theory of convergence, and showing two paramount applications of the theory to the presentation of the Lebesgue measure and the family of Hausdorff measures. The book is largely self-contained, with supplementary sections on topology and differential calculus, and an appendix on filters and ultrafilters also included to help the reader to fully understand the notion of convergence with respect to a filter.
