Details

Autor(en) / Beteiligte

• Stroock, Daniel W
• Stroock, Daniel W

Titel

Essentials of Integration Theory for Analysis [electronic resource]

Auflage

1st ed. 2011

Ort / Verlag

New York, NY : Springer New York

Erscheinungsjahr

2011

Link zum Volltext

• SpringerLink Books - AutoHoldings

Beschreibungen/Notizen

• "A substantial revision of ... [the author's] A concise introduction to the theory of integration ... appropriate as a text for a one-semester graduate course in integration theory"--Back cover.
• Includes bibliographical references and index.
• ""Essentials of Integration Theory for Analysis""; ""Preface""; ""Contents""; ""CHAPTER 1 The Classical Theory""; ""1.1 Riemann Integration""; ""Exercises for  1.1""; ""1.2 Riemannâ€?Stieltjes Integration""; ""1.2.1. Riemann Integrability""; ""1.2.2. Functions of Bounded Variation""; ""Exercises for  1.2""; ""1.3 Rate of Convergence""; ""1.3.1. Periodic Functions""; ""1.3.2. The Non-Periodic Case""; ""Exercises for  1.3""; ""CHAPTER 2 Measures""; ""2.1 Some Generalities""; ""2.1.1. The Idea""; ""2.1.2. Measures and Measure Spaces""; ""Exercises for  2.1""
• ""Exercises for  3.2""""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""; ""Exercises for  3.3""; ""CHAPTER 4 Products of Measures""; ""4.1 Fubini's Theorem""; ""Exercises for  4.1""; ""4.2 Steiner Symmetrization""; ""4.2.1. The Isodiametric inequality""; ""4.2.2. Hausdorff's Description of Lebesgue's Measure""; ""Exercises for  4.2""; ""CHAPTER 5 Changes of Variable""; ""5.1 Riemann vs. Lebesgue, Distributions, and Polar Coordinates""; ""5.1.1. Riemann vs. Lebesgue""
• ""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""; ""Exercises for  6.3""; ""CHAPTER 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""; ""Exercises for  7.1""; ""7.2 Fourier Series""; ""7.2.1. The Fourier Basis""; ""7.2.2. An Application to Eulerâ€?Maclaurin""; ""Exercises for  7.2""; ""7.3 The Fourier Transform""
• ""7.3.1. L¹-Theory of the Fourier Transform""
• Essentials of Integration Theory for Analysis is a substantial revision of the best-selling Birkhäuser title by the same author,  A Concise Introduction to the Theory of Integration. Highlights of this new textbook for the GTM series include revisions to Chapter 1 which add a section about the rate of convergence of Riemann sums and introduces a discussion of the Euler–MacLauren formula.  In Chapter 2, where Lebesque’s theory is introduced, a construction of the countably additive measure is done with sufficient generality to cover both Lebesque and Bernoulli  measures. Chapter 3 includes a proof of Lebesque’s differential theorem for all monotone functions and the concluding chapter has been expanded to include a proof of Carathéory’s  method for constructing measures and his result is applied to the construction of the Hausdorff measures. This new gem is appropriate as a text for a one-semester graduate course in integration theory and is complimented by the addition of several problems related to the new material.  The text is also highly useful for self-study. A complete solutions manual is available for instructors who adopt the text for their courses. Additional publications by Daniel W. Stroock:  An Introduction to Markov Processes,  ©2005 Springer (GTM 230), ISBN: 978-3-540-23499-9; A Concise Introduction to the Theory of Integration, © 1998 Birkhäuser Boston, ISBN: 978-0-8176-4073-6;  (with S.R.S. Varadhan) Multidimensional Diffusion Processes, © 1979 Springer (Classics in Mathematics), ISBN: 978-3-540-28998-2.
• English