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Lebesgue Integration : Universitext - Soo B. Chae

Lebesgue Integration


Paperback Published: 16th December 1994
ISBN: 9780387943572
Number Of Pages: 284

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The fundamental concept of Lebesgue integration provides a basis for approaching theoretical and applied problems in mathematics, engineering, physics and many other sciences.
Lebesgue Integration presents a logical development of the basic concepts of Lebesgue's integration theorems, proceeding from the study of topological concepts on the real line.
This text is intended for advanced undergraduates and beginning graduate students and is also suitable for self-study by students preparing for the masters or doctoral preliminary examination.

Preface to the Second Edition
Preface to the First Edition
Preliminariesp. 1
Setsp. 1
Relationsp. 4
Countable Setsp. 6
Real Numbersp. 8
Topological Concepts in Rp. 10
Continuous Functionsp. 15
Metric Spacesp. 18
The Riemann Integralp. 24
The Cauchy Integralp. 24
Fourier Series and Dirichlet's Conditionsp. 27
The Riemann Integralp. 31
Sets of Measure Zerop. 35
Existence of the Riemann Integralp. 40
Deficiencies of the Riemann Integralp. 44
The Lebesgue Integral: Riesz Methodp. 50
Step Functions and Their Integralsp. 51
Two Fundamental Lemmasp. 55
The Class L[superscript +]p. 58
The Lebesgue Integralp. 63
The Beppo Levi Theorem - Monotone Convergence Theoremp. 67
The Lebesgue Theorem - Dominated Convergence Theoremp. 73
The Space L[superscript +]p. 79
Appendix: Henri Lebesguep. 84
Appendix: Frigyes Rieszp. 85
Lebesgue Measurep. 87
Measurable Functionsp. 87
Lebesgue Measurep. 90
[sigma]-Algebras and Borel Setsp. 94
Nonmeasurable Setsp. 97
Structure of Measurable Setsp. 101
More About Measurable Functionsp. 104
Egoroff's Theoremp. 106
Steinhaus' Theoremp. 112
The Cauchy Functional Equationp. 115
Lebesgue Outer and Inner Measuresp. 118
Generalizationsp. 125
The Integral on Measurable Setsp. 125
The Integral on Infinite Intervalsp. 132
Lebesgue Measure on Rp. 135
Finite Additive Measure: The Banach Measure Problemp. 139
The Double Lebesgue Integral and the Fubini Theoremp. 143
The Complex Integralp. 152
Differentiation and the Fundamental Theorem of Calculusp. 155
Nowhere Differentiable Functionsp. 156
The Dini Derivativesp. 160
The Rising Sun Lemma and Differentiability of Monotone Functionsp. 162
Functions of Bounded Variationp. 171
Absolute Continuityp. 178
The Fundamental Theorem of Calculusp. 184
The L[superscript p] Spaces and the Riesz-Fischer Theoremp. 191
The L[superscript p] Spaces [actual symbol not reproducible]p. 191
Approximations by Continuous Functionsp. 199
The Space [actual symbol not reproducible]p. 205
The l[superscript p] Spaces [actual symbol not reproducible]p. 209
Hilbert Spacesp. 211
The Riesz-Fischer Theoremp. 216
Orthonormalizationp. 224
Completeness of the Trigonometric Systemp. 226
Isoperimetric Problemp. 229
Remarks on Fourier Seriesp. 232
Appendix: The Development of the Notion of the Integralp. 234
Bibliographyp. 249
Notationp. 254
Indexp. 257
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387943572
ISBN-10: 0387943579
Series: Universitext
Audience: General
Format: Paperback
Language: English
Number Of Pages: 284
Published: 16th December 1994
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.32 x 15.62  x 1.78
Weight (kg): 0.4
Edition Number: 2
Edition Type: Revised

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