Lebesgue Integration On Euclidean Space, Revised Edition / Edition 1

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Overview

Lebesgue Integration on Euclidean Space contains a concrete, intuitive, and patient derivation of Lebesgue measure and integration on Rn. Throughout the text, many exercises are incorporated, enabling students to apply new ideas immediately. Jones strives to present a slow introduction to Lebesgue integration by dealing with n-dimensional spaces from the outset. In addition, the text provides students a thorough treatment of Fourier analysis, while holistically preparing students to become workers in real analysis.

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Editorial Reviews

Booknews
The treatment of integration developed by Henri Lebesgue almost a century ago rendered previous theories obsolete and has yet to be replaced by a better one. The author presents an extended introduction to Lebesgue integration, deals with n-dimensional space from the outset, and provides a thorough treatment of Fourier analysis. Other topics include Lebesgue measure, invariance, Cantor sets, algebras of sets and measurable functions, the gamma function, convolutions, and products of abstract measures. Many exercises are incorporated with the text. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780763717087
  • Publisher: Jones & Bartlett Learning
  • Publication date: 11/8/2000
  • Edition description: 1E
  • Edition number: 1
  • Pages: 588

Table of Contents

Preface
Bibliography
Acknowledgments
1Introduction to R[superscript n]1
ASets1
BCountable Sets4
CTopology5
DCompact Sets10
EContinuity15
FThe Distance Function20
2Lebesgue Measure on R[superscript n]25
AConstruction25
BProperties of Lebesgue Measure49
CAppendix: Proof of P1 and P260
3Invariance of Lebesgue Measure65
ASome Linear Algebra66
BTranslation and Dilation71
COrthogonal Matrices73
DThe General Matrix75
4Some Interesting Sets81
AA Nonmeasurable Set81
BA Bevy of Cantor Sets83
CThe Lebesgue Function86
DAppendix: The Modulus of Continuity of the Lebesgue Functions95
5Algebras of Sets and Measurable Functions103
AAlgebras and [sigma]-Algebras103
BBorel Sets107
CA Measurable Set which Is Not a Borel Set110
DMeasurable Functions112
ESimple Functions117
6Integration121
ANonnegative Functions121
BGeneral Measurable Functions130
CAlmost Everywhere135
DIntegration Over Subsets of R[superscript n]139
EGeneralization: Measure Spaces142
FSome Calculations147
GMiscellany152
7Lebesgue Integral on R[superscript n]157
ARiemann Integral157
BLinear Change of Variables170
CApproximation of Functions in L[superscript 1]171
DContinuity of Translation L[superscript 1]180
8Fubini's Theorem for R[superscript n]181
9The Gamma Function199
ADefinition and Simple Properties199
BGeneralization202
CThe Measure of Balls205
DFurther Properties of the Gamma Function209
EStirling's Formula212
FThe Gamma Function on R216
10L[superscript p] Spaces221
ADefinition and Basic Inequalities221
BMetric Spaces and Normed Spaces227
CCompleteness of L[superscript p]231
DThe Case p = [actual symbol not reproducible]235
ERelations between L[superscript p] Spaces238
FApproximation by C[actual symbol not reproducible](R[superscript n])244
GMiscellaneous Problems246
HThe Case 0 < p < 1250
11Products of Abstract Measures255
AProducts of [sigma]-Algebras255
BMonotone Classes258
CConstruction of the Product Measure261
DThe Fubini Theorem268
EThe Generalized Minkowski Inequality271
12Convolutions277
AFormal Properties277
BBasic Inequalities280
CApproximate Identities284
13Fourier Transform of R[superscript n]293
AFourier Transform of Functions in L[superscript 1](R[superscript n])293
BThe Inversion Theorem308
CThe Schwartz Class320
DThe Fourier-Plancherel Transform323
EHilbert Space334
FFormal Applications to Differential Equations339
GBessel Functions344
HSpecial Results for n = 1352
IHermite Polynomials356
14Fourier Series in One Variable367
APeriodic Functions367
BTrigonometric Series373
CFourier Coefficients392
DConvergence of Fourier Series400
ESummability of Fourier Series410
FA Counterexample418
GParseval's Identity421
HPoisson Summation Formula428
IA Special Class of Sine Series436
15Differentiation447
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