Integration and Modern Analysis

1. Front Matter

   Pages i-xix

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2. Classical Real Variables

   • John J. Benedetto, Wojciech Czaja

   Pages 1-35

3. Lebesgue Measure and General Measure Theory

   • John J. Benedetto, Wojciech Czaja

   Pages 37-94

4. The Lebesgue Integral

   • John J. Benedetto, Wojciech Czaja

   Pages 95-164

5. The Relationship between Differentiation and Integration on \( \mathbb{R}\)

   • John J. Benedetto, Wojciech Czaja

   Pages 165-218

6. Spaces of Measures and the Radon–Nikodym Theorem

   • John J. Benedetto, Wojciech Czaja

   Pages 219-276

7. Weak Convergence of Measures

   • John J. Benedetto, Wojciech Czaja

   Pages 277-320

8. Riesz Representation Theorem

   • John J. Benedetto, Wojciech Czaja

   Pages 321-357

9. Lebesgue Differentiation Theorem on \( \mathbb{R}^d \)

   • John J. Benedetto, Wojciech Czaja

   Pages 359-406

10. Self-Similar Sets and Fractals

    • John J. Benedetto, Wojciech Czaja

    Pages 407-439

11. Functional Analysis

    • John J. Benedetto, Wojciech Czaja

    Pages 441-485

12. Fourier Analysis

    • John J. Benedetto, Wojciech Czaja

    Pages 487-528

13. Back Matter

    Pages 529-575

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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.

• Complex variables
• Distribution
• Lebesgue differentiation theorem
• Maximum
• Measure theory
• Real analysis
• Riesz representation theorem
• calculus
• differential equation
• functional analysis
• measure

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 andModern 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–Nikodým 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, Universität 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 bothinstructive and agreeable. It can be recommended … to students for self-study and to researchers in various domains of analysis as a reference text.” (S. Cobzaş, Studia Universitatis Babes-Bolyai, Mathematica, Vol. LV (4), December, 2010)

• Dept. Mathematics, University of Maryland, College Park, U.S.A.

  John J. Benedetto, Wojciech Czaja

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