9780691113869

Real Analysis : Measure Theory, Integration, and Hilbert Spaces

by
  • ISBN13:

    9780691113869

  • ISBN10:

    0691113866

  • Format: Hardcover
  • Copyright: 3/14/2005
  • Publisher: Princeton Univ Pr

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Summary

Real Analysisis the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science. After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises. As with the other volumes in the series,Real Analysisis accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels. Also available, the first two volumes in the Princeton Lectures in Analysis:

Table of Contents

Foreword vii
Introduction xv
1 Fourier series: completion
xvi
2 Limits of continuous functions
xvi
3 Length of curves
xvii
4 Differentiation and integration
xviii
5 The problem of measure
xviii
Chapter 1. Measure Theory 1(48)
1 Preliminaries
1(9)
2 The exterior measure
10(6)
3 Measurable sets and the Lebesgue measure
16(11)
4 Measurable functions
27(7)
4.1 Definition and basic properties
27(3)
4.2 Approximation by simple functions or step functions
30(3)
4.3 Littlewood's three principles
33(1)
5 The Brunn-Minkowski inequality
34(3)
6 Exercises
37(9)
7 Problems
46(3)
Chapter 2. Integration Theory 49(49)
1 The Lebesgue integral: basic properties and convergence theorems
49(19)
2 The space L¹ of integrable functions
68(7)
3 Fubini's theorem
75(11)
3.1 Statement and proof of the theorem
75(5)
3.2 Applications of Fubini's theorem
80(6)
4 A Fourier inversion formula
86(3)
5 Exercises
89(6)
6 Problems
95(3)
Chapter 3. Differentiation and Integration 98(58)
1 Differentiation of the integral
99(9)
1.1 The Hardy-Littlewood maximal function
100(4)
1.2 The Lebesgue differentiation theorem
104(4)
2 Good kernels and approximations to the identity
108(6)
3 Differentiability of functions
114(20)
3.1 Functions of bounded variation
115(12)
3.2 Absolutely continuous functions
127(4)
3.3 Differentiability of jump functions
131(3)
4 Rectifiable curves and the isoperimetric inequality
134(11)
4.1 Minkowski content of a curve
136(7)
4.2 Isoperimetric inequality
143(2)
5 Exercises
145(7)
6 Problems
152(4)
Chapter 4. Hilbert Spaces: An Introduction 156(51)
1 The Hilbert space L²
156(5)
2 Hilbert spaces
161(9)
2.1 Orthogonality
164(4)
2.2 Unitary mappings
168(1)
2.3 Pre-Hilbert spaces
169(1)
3 Fourier series and Fatou's theorem
170(4)
3.1 Fatou's theorem
173(1)
4 Closed subspaces and orthogonal projections
174(6)
5 Linear transformations
180(8)
5.1 Linear functionals and the Riesz representation theorem
181(2)
5.2 Adjoints
183(2)
5.3 Examples
185(3)
6 Compact operators
188(5)
7 Exercises
193(9)
8 Problems
202(5)
Chapter 5. Hilbert Spaces: Several Examples 207(55)
1 The Fourier transform on L2
207(6)
2 The Hardy space of the upper half-plane
213(8)
3 Constant coefficient partial differential equations
221(8)
3.1 Weak solutions
222(2)
3.2 The main theorem and key estimate
224(5)
4 The Dirichlet principle
229(24)
4.1 Harmonic functions
234(9)
4.2 The boundary value problem and Dirichlet's principle
243(10)
5 Exercises
253(6)
6 Problems
259(3)
Chapter 6. Abstract Measure and Integration Theory 262(61)
1 Abstract measure spaces
263(10)
1.1 Exterior measures and Carathéodory's theorem
264(2)
1.2 Metric exterior measures
266(4)
1.3 The extension theorem
270(3)
2 Integration on a measure space
273(3)
3 Examples
276(9)
3.1 Product measures and a general Fubini theorem
276(3)
3.2 Integration formula for polar coordinates
279(2)
3.3 Borel measures on R and the Lebesgue-Stieltjes integral
281(4)
4 Absolute continuity of measures
285(7)
4.1 Signed measures
285(3)
4.2 Absolute continuity
288(4)
5 Ergodic theorems
292(14)
5.1 Mean ergodic theorem
294(2)
5.2 Maximal ergodic theorem
296(4)
5.3 Pointwise ergodic theorem
300(2)
5.4 Ergodic measure-preserving transformations
302(4)
6 Appendix: the spectral theorem
306(6)
6.1 Statement of the theorem
306(1)
6.2 Positive operators
307(2)
6.3 Proof of the theorem
309(2)
6.4 Spectrum
311(1)
7 Exercises
312(7)
8 Problems
319(4)
Chapter 7. Hausdorff Measure and Fractals 323(66)
1 Hausdorff measure
324(5)
2 Hausdorff dimension
329(20)
2.1 Examples
330(11)
2.2 Self-similarity
341(8)
3 Space-filling curves
349(11)
3.1 Quartic intervals and dyadic squares
351(2)
3.2 Dyadic correspondence
353(2)
3.3 Construction of the Peano mapping
355(5)
4 Besicovitch sets and regularity
360(20)
4.1 The Radon transform
363(7)
4.2 Regularity of sets when d > or equal to 3
370(1)
4.3 Besicovitch sets have dimension 2
371(3)
4.4 Construction of a Besicovitch set
374(6)
5 Exercises
380(5)
6 Problems
385(4)
Notes and References 389(2)
Bibliography 391(4)
Symbol Glossary 395(2)
Index 397

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