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**About this book :- **
**Measure, Integration & Real Analysis ** written by
** Sheldon Axler **

This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics. Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn.

Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability.

Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that is freely available online.

The basic prerequisite for your students to use this textbook is a good understanding of elementary undergraduate real analysis. Your students can download from the book’s website (http://measure.axler.net) or from the Springer website the document titled Supplement for Measure, Integration & Real Analysis. That supplement can serve as a review of the elementary undergraduate real analysis used in this book.

(Sheldon Axler)

**Book Detail :- **
** Title: ** Measure, Integration & Real Analysis
** Edition: **
** Author(s): ** Sheldon Axler
** Publisher: ** Springer International Publishing
** Series: ** Graduate Texts in Mathematics 282
** Year: ** 2020
** Pages: ** 430
** Type: ** PDF
** Language: ** English
** ISBN: ** 978-3-030-33142-9,978-3-030-33143-6
** Country: ** US

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**About Author :- **

Author ** Sheldon Axler ** was valedictorian of his high school in Miami, Florida. He received his AB from Princeton University with highest honors, followed by a PhD in Mathematics from the University of California at Berkeley. As a Moore Instructor at MIT, Axler received a university-wide teaching award.

Axler was then an assistant professor, associate professor, and professor in the Mathematics Department at Michigan State University, where he received the first J. Sutherland Frame Teaching Award and the Distinguished Faculty Award.

Axler received the Lester R. Ford Award for expository writing from the Mathematical Association of America in 1996. In addition to publishing numerous research papers, Axler is the author of five mathematics textbooks, ranging from freshman to graduate level. His book Linear Algebra Done Right has been adopted as a textbook at over 260 universities.

Axler has served as Editor-in-Chief of the Mathematical Intelligencer and as Associate Editor of the American Mathematical Monthly. He has been a member of the Council of the American Mathematical Society and a member of the Board of Trustees of the Mathematical Sciences Research Institute. Axler currently serves on the editorial board of Springer’s series Undergraduate Texts in Mathematics, Graduate Texts in Mathematics, and Universitext.

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**Book Contents :- **
**Measure, Integration & Real Analysis ** written by
** Sheldon Axler **
cover the following topics.

Acknowledgments

1. Riemann Integration

1. A Review: Riemann Integral

1. B Riemann Integral Is Not Good Enough

2. Measures

2. A Outer Measure on R

2. B Measurable Spaces and Functions

2. C Measures and Their Properties

2. D Lebesgue Measure

2. E Convergence of Measurable Functions

3. Integration

3. A Integration with Respect to a Measure

3. B Limits of Integrals & Integrals of Limits

4. Differentiation

4. A Hardy–Littlewood Maximal Function

4. B Derivatives of Integrals

5. Product Measures

5. A Products of Measure Spaces

5. B Iterated Integrals

5. C Lebesgue Integration on Rn

6. Banach Spaces

6. A Metric Spaces

6. B Vector Spaces

6. C Normed Vector Spaces

6. D Linear Functionals

6. E Consequences of Baire’s Theorem

7. Lp Spaces

7. A Lp(m)

7. B Lp(m)

8. Hilbert Spaces

8. A Inner Product Spaces

8. B Orthogonality

8. C Orthonormal Bases

9. Real and Complex Measures

9. A Total Variation

9. B Decomposition Theorems

10. Linear Maps on Hilbert Spaces

10. A Adjoints and Invertibility

10. B Spectrum

10. C Compact Operators

10. D Spectral Theorem for Compact Operators

11. Fourier Analysis

11. A Fourier Series and Poisson Integral

11. B Fourier Series and Lp of Unit Circle

11. C Fourier Transform

12. Probability Measures

Photo Credits

Bibliography

Notation Index

Index

Colophon: Notes on Typesetting

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- Abstract Algebra
- Calculus
- Differential Equations
- Engineering Mathematics
- Linear Algebra
- Math Magic
- Real Analysis