When the first edition of this textbook published in 2011, it constituted a substantial revision of the best-selling Birkhäuser title by the same author, A Concise Introduction to the Theory of Integration. Appropriate as a primary text for a one-semester graduate course in integration theory, this GTM is also useful for independent study. A complete solutions manual is available for instructors who adopt the text for their courses. This second edition has been revised as follows: §2.2.5 and §8.3 have been substantially reworked. New topics have been added. As an application of the material about Hermite functions in §7.3.2, the author has added a brief introduction to Schwartz's theory of tempered distributions in §7.3.4. Section §7.4 is entirely new and contains applications, including the Central Limit Theorem, of Fourier analysis to measures. Related to this are subsections §8.2.5 and §8.2.6, where Lévy's Continuity Theorem and Bochner's characterization of the Fourier transforms of Borel probability on ℝN are proven. Subsection 8.1.2 is new and contains a proof of the Hahn Decomposition Theorem. Finally, there are several new exercises, some covering material from the original edition and others based on newly added material.
Author(s): Daniel W. Stroock
Series: Graduate Texts in Mathematics
Edition: 2
Publisher: Springer
Year: 2020
Language: English
Pages: 285
Tags: Measure Theory, Lebesgue Integral, Lebesgue Spaces, Fourier Analysis
Preface
Contents
Notation
1 The Classical Theory
1.1 Riemann Integration
1.1.1 Exercises for 1671.1
1.2 Riemann–Stieltjes Integration
1.2.1 Riemann–Stieltjes Integrability
1.2.2 Functions of Bounded Variation
1.2.3 Exercises for 1671.2
1.3 Rate of Convergence
1.3.1 Periodic Functions
1.3.2 The Non-Periodic Case
1.3.3 Exercises for 1671.3
2 Measures
2.1 Some Generalities
2.1.1 The Idea
2.1.2 Measures and Measure Spaces
2.1.3 Exercises for 1672.1
2.2 A Construction of Measures
2.2.1 A Construction Procedure
2.2.2 Lebesgue Measure on mathbbRN
2.2.3 Distribution Functions and Measures
2.2.4 Bernoulli Measure
2.2.5 Bernoulli and Lebesgue Measures
2.2.6 Exercises for 1672.2
3 Lebesgue Integration
3.1 The Lebesgue Integral
3.1.1 Some Miscellaneous Preliminaries
3.1.2 The Space L1(µ;mathbbR)
3.1.3 Exercises for 1673.1
3.2 Convergence of Integrals
3.2.1 The Big Three Convergence Results
3.2.2 Convergence in Measure
3.2.3 Elementary Properties of L1(µ;mathbbR)
3.2.4 Exercises for 1673.2
3.3 Lebesgue's Differentiation Theorem
3.3.1 The Sunrise Lemma
3.3.2 The Absolutely Continuous Case
3.3.3 The General Case
3.3.4 Exercises for 1673.3
4 Products of Measures
4.1 Fubini's Theorem
4.1.1 Exercises for 1674.1
4.2 Steiner Symmetrization
4.2.1 The Isodiametric inequality
4.2.2 Hausdorff's Description of Lebesgue's Measure
4.2.3 Exercises for 1674.2
5 Changes of Variable
5.1 Riemann vs. Lebesgue, Distributions, and Polar Coordinates
5.1.1 Riemann vs. Lebesgue
5.1.2 Polar Coordinates
5.1.3 Exercises for 1675.1
5.2 Jacobi's Transformation and Surface Measure
5.2.1 Jacobi's Transformation Formula
5.2.2 Surface Measure
5.2.3 Exercises for 1675.2
5.3 The Divergence Theorem
5.3.1 Flows Generated by Vector Fields
5.3.2 Mass Transport
5.3.3 Exercises for 1675.3
6 Basic Inequalities and Lebesgue Spaces
6.1 Jensen, Minkowski, and Hölder
6.1.1 Exercises for 1676.1
6.2 The Lebesgue Spaces
6.2.1 The Lp-Spaces
6.2.2 Mixed Lebesgue Spaces
6.2.3 Exercises for 1676.2
6.3 Some Elementary Transformations on Lebesgue Spaces
6.3.1 A General Estimate for Linear Transformations
6.3.2 Convolutions and Young's inequality
6.3.3 Friedrichs Mollifiers
6.3.4 Exercises for 1676.3
7 Hilbert Space and Elements of Fourier Analysis
7.1 Hilbert Space
7.1.1 Elementary Theory of Hilbert Spaces
7.1.2 Orthogonal Projection and Bases
7.1.3 Exercises for 1677.1
7.2 Fourier Series
7.2.1 The Fourier Basis
7.2.2 An Application to Euler–Maclaurin
7.2.3 Exercises for 1677.2
7.3 The Fourier Transform
7.3.1 L1-Theory of the Fourier Transform
7.3.2 The Hermite Functions
7.3.3 L2-Theory of the Fourier Transform
7.3.4 Schwartz Test Function Space and Tempered Distributions
7.3.5 Exercises for 1677.3
7.4 The Fourier Transform of Probability Measures
7.4.1 Parseval for Measures
7.4.2 Weak Convergence of Probability Measures
7.4.3 Convolutions and The Central Limit Theorem
7.4.4 Exercises for 1677.4
8 Radon–Nikodym, Hahn, Daniell, and Carathéodory
8.1 The Radon–Nikodym and Hahn Decomposition Theorems
8.1.1 The Radon-Nikodym Theorem
8.1.2 Hahn Decomposition
8.1.3 Exercises for 1678.1
8.2 The Daniell Integral
8.2.1 Extending an Integration Theory
8.2.2 Identification of the Measure
8.2.3 An Extension Theorem
8.2.4 Another Riesz Representation Theorem
8.2.5 Lévy Continuity and Bochner's Theorems
8.2.6 Exercises for 1678.2
8.3 Carathéodory's Method
8.3.1 Outer Measures and Measurability
8.3.2 Carathéodory's Criterion
8.3.3 Hausdorff Measures
8.3.4 Hausdorff Measure and Surface Measure
8.3.5 Exercises for 1678.3
Index
