This book is both a text and a paean to twentieth-century real variables, measure theory, and integration theory. As a text, the book is aimed at graduate students. As an exposition, extolling this area of analysis, the book is necessarily limited in scope and perhaps unnecessarily unlimited in id- syncrasy. More than half of this book is a fundamental graduate real variables course as we now teach it. Since there are excellent textbooks that generally cover the course material herein, part of this Preface renders an apologia for our content, presentation, and existence. The following section presents our syllabus properly liberated from too many demands. Subsequent sections dealwithoutline,theme,features,andtherolesofFourieranalysisandVitali, respectively. Mathematics is a creativeadventure drivenby beauty, structure, intrinsic mathematicalproblems,extrinsicproblemsfromengineeringandthesciences, and serendipity. This book treats integration theory and its fascinating creation through the past century. What about the rest of our title? "Analysis"is many subjects to many mathematicians."Modern analysis" is hardly a constraint for a single volume such as ours; one can argue the opposite. Di?erentiation and integrationare still the essence of analysis,and, along with "integration", the title could very well have included the word "di?erentiation" because of our emphasis on it. Guided by the creativity of mathematics,ourtitleismeanttoassertthatthetechnologywehaverecorded is a basis for many of the analytic adventures of our time. Syllabus We shall outline the material we have used in teaching a ?rst-year graduate course in real analysis. Sometimes a student will take only the ?rst semester of this two-semester sequence.

A paean to twentieth century analysis, this modern text has several important themes and key features which set it apart from others on the subject. A major thread throughout is the unifying influence of the concept of absolute continuity on differentiation and integration. This leads to fundamental results such as the Dieudonné-Grothendieck theorem and other intricate developments dealing with weak convergence of measures.

Key Features:

* Fascinating historical commentary interwoven into the exposition;

* Hundreds of problems from routine to challenging;

* Broad mathematical perspectives and material, e.g., in harmonic analysis and probability theory, for independent study projects;

* Two significant appendices on functional analysis and Fourier analysis.

Key Topics:

* In-depth development of measure theory and Lebesgue integration;

* Comprehensive treatment of connection between differentiation and integration, as well as complete proofs of state-of-the-art results;

* Classical real variables and introduction to the role of Cantor sets, later placed in the modern setting of self-similarity and fractals;

* Evolution of the Riesz representation theorem to Radon measures and distribution theory;

* Deep results in modern differentiation theory;

* Systematic development of weak sequential convergence inspired by theorems of Vitali, Nikodym, and Hahn-Saks;

* Thorough treatment of rearrangements and maximal functions;

* The relation between surface measure and Hausforff measure;

* Complete presentation of Besicovich coverings and differentiation of measures.

Integration and Modern Analysis will serve advanced undergraduates and graduate students, as well as professional mathematicians. It may be used in the classroom or self-study.

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