Introduction to modern analysis

447 Pages·2003·1.906 MB·English

by  Shmuel Kantorovitz

Oxford Graduate Texts in Mathematics
Series Editors
R. Cohen S. K. Donaldson
S. Hildebrandt T. J. Lyons
M. J. Taylor
oxford graduate texts in mathematics
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2. Reinhold Meise and Dietmar Vogt: Introduction to Functional Analysis
3. James G. Oxley: Matroid Theory
4. N. J. Hitchin, G. B. Segal, and R. S. Ward: Integrable Systems: Twistors,
Loop Groups, and Riemann Surfaces
5. Wulf Rossmann: Lie Groups: An Introduction Through Linear Groups
6. Q. Liu: Algebraic Geometry and Arithmetic Curves
7. Martin R. Bridson and Simon M, Salamon (eds): Invitations to Geometry
and Topology
8. Shmuel Kantorovitz: Introduction to Modern Analysis
9. Terry Lawson: Topology: A Geometric Approach
10. Meinolf Geck: An Introduction to Algebraic Geometry and Algebraic
Groups
Introduction to
Modern Analysis
Shmuel Kantorovitz
Bar Ilan University,
Ramat Gan, Israel
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ISBN 0–19–852656–3 978–0–19–852656–8
ISBN 0–19–920315–6(Pbk.) 978–0–19–920315–4(Pbk.)
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To Ita, Bracha, Pnina, Pinchas, and Ruth
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Preface
This book grew out of lectures given since 1964 at Yale University, the Uni-
versity of Illinois at Chicago, and Bar Ilan University. The material covers the
usual topics of Measure Theory and Functional Analysis, with applications to
Probability Theory and to the theory of linear partial differential equations.
Somerelativelyadvancedtopicsareincludedineachchapter(excludingthefirst
two):theRiesz–MarkovrepresentationtheoremanddifferentiabilityinEuclidean
spaces (Chapter 3); Haar measure (Chapter 4); Marcinkiewicz’s interpola-
tion theorem (Chapter 5); the Gelfand–Naimark–Segal representation theorem
(Chapter 7); the Von Neumann double commutant theorem (Chapter 8); the
spectralrepresentationtheoremfornormaloperators(Chapter9); theextension
theoryforunboundedsymmetricoperators(Chapter10);theLyapounovCentral
Limit theorem and the Kolmogoroff ‘Three Series theorem’ (Application I); the
Hormander–Malgrangetheorem,fundamentalsolutionsoflinearpartialdifferen-
tial equations with variable coefficients, and Hormander’s theory of convolution
operators, with an application to integration of pure imaginary order (Applica-
tion II). Some important complementary material is included in the ‘Exercises’
sections,withdetailedhintsleadingstep-by-steptothewantedresults.Solutions
to the end of chapter exercises may be found on the companion website for this
text: http://www.oup.co.uk/academic/companion/mathematics/kantorovitz.
Ramat Gan S. K.
July 2002
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Contents
1 Measures 1
1.1 Measurable sets and functions 1
1.2 Positive measures 7
1.3 Integration of non-negative measurable functions 9
1.4 Integrable functions 15
1.5 Lp-spaces 22
1.6 Inner product 29
1.7 Hilbert space: a first look 32
1.8 The Lebesgue–Radon–Nikodym theorem 34
1.9 Complex measures 39
1.10 Convergence 46
1.11 Convergence on finite measure space 49
1.12 Distribution function 50
1.13 Truncation 52
Exercises 54
2 Construction of measures 57
2.1 Semi-algebras 57
2.2 Outer measures 59
2.3 Extension of measures on algebras 62
2.4 Structure of measurable sets 63
2.5 Construction of Lebesgue–Stieltjes measures 64
2.6 Riemann versus Lebesgue 67
2.7 Product measure 69
Exercises 73
3 Measure and topology 77
3.1 Partition of unity 77
3.2 Positive linear functionals 79
3.3 The Riesz–Markov representation theorem 87
3.4 Lusin’s theorem 89
3.5 The support of a measure 92
3.6 Measures on Rk; differentiability 93
Exercises 97