Integration and Modern Analysis: From Classical Real Variables to Distribution Theory (Birkhäuser Advanced Texts Basler Lehrbücher)

From the reviews:

This is one of the best graduate texts on real variables, measure theory and integration theory known to the reviewer. ... The book is exceptionally accurate in up-to-date contents, structure of the text, comprehensibility and references (which is an excellent source for professionals). ... In addition to the numerous illustrative examples and applications, [Integration and Modern Analysis] is replete with interesting remarks and historical notes contained either in the body of the text or in the `Potpurri and titillation' section at the end of each chapter. ... A subject Index, Index of Names, Index of Notation and a carefully prepared preface accompany this volume, being a useful guide to any reader. For teachers and for researchers, the book will prove a priceless resource, an avenue to new and often surprising ideas. Physically beautiful and elegantly printed, organized and written lucidly, this volume is a welcome addition and an important contribution to mathematical literature. -Mathematical Reviews (Grigore Ciurea, Romania)

Graduate analysis books are as common as tea served at department colloquia! Therefore, Integration and Modern Analysis by Benedetto and Czaja had better have something special to distinguish itself...Integration and Modern Analysis is not just your average cup of tea. Its goals go well beyond the usual prosaic objective of presenting rookie graduate students with a certain standard set of tools and skills in real analysis; Benedetto and Czaja aim to persuade the reader to their particular point of view and, indeed, to enlist him in their enterprise...Well, Benedetto and Czaja have a wonderful product to sell and are right in doing such enthusiastic preaching of their cause. Additionally, the exposition is solid, the book is loaded with exercises, and is dripping with the authors' expertise. If you incline in this direction of analysis, Integration and Modern Analysis is unquestionably your cup of tea. -MAA Reviews (Michael Berg, Loyola Marymount University, Los Angeles, CA)

For the authors, the notion of 'absolute continuity', tracing back to Vitali, is basic for the whole theory as the unifying concept for all major results, such as the fundamental theorem of calculus, the Lebesgue dominated convergence theorem, and the Radon-Nikodym theorem. They find, that as yet in no textbook this point of view was carried through in an adequate way. Their text is intended as a remedy to the latter. As a consequence they include very carefully materials which are often omitted in monographs about real analysis and integration theory, such as Vitali's necessary and sufficient conditions for interchanging limits and integrals, the Vitali-Hahn-Saks theorem, but also, quite unusually, Grothendieck's theory of weak convergence of measures. The presentation of the material becomes most vivid by numerous remarks concerning the historical genesis, by biographical notes on the most important contributors, especially about Vitali, by discussion of problems, and by an excellent choice of examples and comments for motivation to an extent, that seem to be unique...All chapters end with sections (`potpourri and titillations') devoted to relations with other fields and problems intended for presenting new perspectives. This way a lot of information is provided and this makes the book interesting also for advanced study and for anybody interested in the field. -Zentralblatt MATH (Werner Strauss, Universitat Stuttgart)

The aim of the present book is to emphasize how modern integration theory evolved from some classical problems in function theory, related mainly to Fourier analysis. ... a nice book, containing a lot of results in measure theory and integration theory, making good connections between classical and modern ones. The lively style of exposition makes the reading both instructive and agreeable. It can be recommended ... to students for self-study and to researchers in various domains of analysis as a reference text. (S. Cobzas, Studia Universitatis Babes-Bolyai, Mathematica, Vol. LV (4), December, 2010)