A Course on Integration Theory

1. Front Matter

   Pages i-xviii

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2. General Theory of Integration

   • Nicolas Lerner

   Pages 1-66

3. Actual Construction of Measure Spaces

   • Nicolas Lerner

   Pages 67-123

4. Spaces of Integrable Functions

   • Nicolas Lerner

   Pages 125-187

5. Integration on a Product Space

   • Nicolas Lerner

   Pages 189-217

6. Diffeomorphisms of Open Subsets of ℝⁿ and Integration

   • Nicolas Lerner

   Pages 219-281

7. Convolution

   • Nicolas Lerner

   Pages 283-316

8. Complex Measures

   • Nicolas Lerner

   Pages 317-341

9. Basic Harmonic Analysis on ℝⁿ

   • Nicolas Lerner

   Pages 343-370

10. Classical Inequalities

    • Nicolas Lerner

    Pages 371-406

11. Appendix

    • Nicolas Lerner

    Pages 407-479

12. Back Matter

    Pages 481-492

    PDF

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.​  

• Fourier transformation
• L^p spaces
• Lebesgue measure
• measure theory

“It is well written and the proofs are given in great detail, so that it can serve as a textbook for students as well as a reference for more advanced readers. It consists of nine chapters and an appendix devoted to making the book as self-contained as possible.” (José Rodríguez, Mathematical Reviews, October, 2016)

• Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie (Paris VI), Paris, France

  Nicolas Lerner

Nicolas Lerner is Professor at Université Pierre and Marie Curie in Paris, France. He held professorial positions in the United States (Purdue University), and in France. His research work is concerned with microlocal analysis and partial differential equations. His recent book Metrics on the Phase Space and Non-Selfadjoint Pseudodifferential Operators was published by Birkhäuser. He was an invited section speaker at the Beijing International Congress of Mathematicians in 2002.

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