**Graduate Studies in Mathematics**

Volume: 45;
2002;
424 pp;
Hardcover

MSC: Primary 28;
Secondary 26

Print ISBN: 978-0-8218-2974-5

Product Code: GSM/45

List Price: $72.00

AMS Member Price: $57.60

MAA member Price: $64.80

**Electronic ISBN: 978-1-4704-2096-3
Product Code: GSM/45.E**

List Price: $72.00

AMS Member Price: $57.60

MAA member Price: $64.80

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# An Introduction to Measure and Integration: Second Edition

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*Inder K. Rana*

Integration is one of the two cornerstones of analysis. Since the fundamental
work of Lebesgue, integration has been interpreted in terms of measure theory.
This introductory text starts with the historical development of the notion of
the integral and a review of the Riemann integral. From here, the reader is
naturally led to the consideration of the Lebesgue integral, where abstract
integration is developed via measure theory. The important basic topics are all
covered: the Fundamental Theorem of Calculus, Fubini's Theorem, \(L_p\)
spaces, the Radon-Nikodym Theorem, change of variables formulas, and so on.

The book is written in an informal style to make the subject matter easily
accessible. Concepts are developed with the help of motivating examples,
probing questions, and many exercises. It would be suitable as a textbook for
an introductory course on the topic or for self-study.

For this edition, more exercises and four appendices have been added.

The AMS maintains exclusive distribution rights for this edition in North America and nonexclusive distribution rights worldwide, excluding India, Pakistan, Bangladesh, Nepal, Bhutan, Sikkim, and Sri Lanka.

#### Readership

Graduate students and research mathematicians interested in mathematical analysis.

#### Reviews & Endorsements

Distinctive features include: 1) An unusually extensive treatment of the historical developments leading up to the Lebesgue integral … 2) Presentation of the standard extension of an abstract measure on an algebra to a sigma algebra prior to the final stage of development of Lebesgue measure. 3) Extensive treatment of change of variables theorems for functions of one and several variables … the conversational tone and helpful insights make this a useful introduction to the topic … The material is presented with generous details and helpful examples at a level suitable for an introductory course or for self-study.

-- Zentralblatt MATH

A special feature [of the book] is the extensive historical and motivational discussion … At every step, whenever a new concept is introduced, the author takes pains to explain how the concept can be seen to arise naturally … The book attempts to be comprehensive and largely succeeds … The text can be used for either a one-semester or a one-year course at M.Sc. level … The book is clearly a labor of love. The exuberance of detail, the wealth of examples and the evident delight in discussing variations and counter examples, all attest to that … All in all, the book is highly recommended to serious and demanding students.

-- Resonance — journal of science education

#### Table of Contents

# Table of Contents

## An Introduction to Measure and Integration: Second Edition

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface xi12 free
- Preface to the Second Edition xvii18 free
- Recipe for a one semester course and interdependence of the chapters xix20 free
- Notations used in the text xxi22 free
- Prologue. The length function 124 free
- Chapter 1. Riemann integration 528
- Chapter 2. Recipes for extending the Riemann integral 4568
- Chapter 3. General extension theory 5174
- §3.1. First extension 5174
- §3.2. Semi-algebra and algebra of sets 5477
- §3.3. Extension from semi-algebra to the generated algebra 5881
- §3.4. Impossibility of extending the length function to all subsets of the real line 6184
- §3.5. Countably additive set functions on intervals 6285
- §3.6. Countably additive set functions on algebras 6487
- §3.7. The induced outer measure 7093
- §3.8. Choosing nice sets: Measurable sets 7497
- §3.9. The σ-algebras and extension from the algebra to the generated σ-algebra 80103
- §3.10. Uniqueness of the extension 84107
- §3.11. Completion of a measure space 89112

- Chapter 4. The Lebesgue measure on R and its properties 95118
- §4.1. The Lebesgue measure 95118
- §4.2. Relation of Lebesgue measurable sets with topologically nice subsets of R 99122
- §4.3. Properties of the Lebesgue measure with respect to the group structure on R 103126
- §4.4. Uniqueness of the Lebesgue measure 106129
- §4.5. *Cardinalities of the σ-algebras L and B[sub(R)] 110133
- §4.6. Nonmeasurable subsets of R 113136
- §4.7. The Lebesgue-Stieltjes measure 114137

- Chapter 5. Integration 117140
- §5.1. Integral of nonnegative simple measurable functions 118141
- §5.2. Integral of nonnegative measurable functions 122145
- §5.3. Intrinsic characterization of nonnegative measurable functions 130153
- §5.4. Integrable functions 143166
- §5.5. The Lebesgue integral and its relation with the Riemann integral 153176
- §5.6. L[sub(1)][a,b] as completion of R[a,b] 158181
- §5.7. Another dense subspace of L[sub(1)][a,b] 163186
- §5.8. Improper Riemann integral and its relation with the Lebesgue integral 168191
- §5.9. Calculation of some improper Riemann integrals 172195

- Chapter 6. Fundamental theorem of calculus for the Lebesgue integral 175198
- Chapter 7. Measure and integration on product spaces 209232
- Chapter 8. Modes of convergence and L[sub(p)]-spaces 243266
- §8.1. Integration of complex-valued functions 243266
- §8.2. Convergence: Pointwise, almost everywhere, uniform and almost uniform 248271
- §8.3. Convergence in measure 255278
- §8.4. L[sub(p)]-space 261284
- §8.5. *Necessary and sufficient conditions for convergence in L[sub(p)] 270293
- §8.6. Dense subspaces of L[sub(p)] 279302
- §8.7. Convolution and regularization of functions 281304
- §8.8. L[sub(∞)](X,S,μ): The space of essentially bounded functions 291314
- §8.9. L[sub(2)](X,S,μ): The space of square integrable functions 296319
- §8.10. L[sub(2)]-convergence of Fourier series 306329

- Chapter 9. The Radon-Nikodym theorem and its applications 311334
- Chapter 10. Signed measures and complex measures 345368
- Appendix A. Extended real numbers 385408
- Appendix B. Axiom of choice 389412
- Appendix C. Continuum hypothesis 391414
- Appendix D. Urysohn's lemma 393416
- Appendix E. Singular value decomposition of a matrix 395418
- Appendix F. Functions of bounded variation 397420
- Appendix G. Differentiable transformations 401424
- References 409432
- Index 413436
- Index of notations 419442 free
- Back Cover Back Cover1450