Real Analysis

Theory Of Measure And Integration (2nd Edition)

By: James J Yeh

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Paperback | 29 June 2006 | Edition Number 2

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762 Pages

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Revised

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22.86 x 15.24 x 3.86

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This book presents a unified treatise of the theory of measure and integration. In the setting of a general measure space, every concept is defined precisely and every theorem is presented with a clear and complete proof with all the relevant details. Counter-examples are provided to show that certain conditions in the hypothesis of a theorem cannot be simply dropped.The dependence of a theorem on earlier theorems is explicitly indicated in the proof, not only to facilitate reading but also to delineate the structure of the theory. The precision and clarity of presentation make the book an ideal textbook for a graduate course in real analysis while the wealth of topics treated also make the book a valuable reference work for mathematicians.

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Published: 29th June 2006

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Preface to the First Edition	p. xiii
Preface to the Second Edition	p. xvii
Notations	p. xix
Measure Spaces	p. 1
Introduction	p. 1
Measure on a [sigma]-algebra of Sets	p. 3
[sigma]-algebra of Sets	p. 3
Limits of Sequences of Sets	p. 4
Generation of [sigma]-algebras	p. 6
Borel [sigma]-algebras	p. 9
Measure on a [sigma]-algebra	p. 11
Measures of a Sequence of Sets	p. 14
Measurable Space and Measure Space	p. 17
Measurable Mapping	p. 19
Induction of Measure by Measurable Mapping	p. 22
Outer Measures	p. 28
Construction of Measure by Means of Outer Measure	p. 28
Regular Outer Measures	p. 32
Metric Outer Measures	p. 34
Construction of Outer Measures	p. 37
Lebesgue Measure on R	p. 41
Lebesgue Outer Measure on R	p. 41
Some Properties of the Lebesgue Measure Space	p. 45
Existence of Non Lebesgue Measurable Sets	p. 49
Regularity of Lebesgue Outer Measure	p. 51
Lebesgue Inner Measure on R	p. 57
Measurable Functions	p. 70
Measurability of Functions	p. 70
Operations with Measurable Functions	p. 74
Equality Almost Everywhere	p. 78
Sequence of Measurable Functions	p. 79
Continuity and Borel and Lebesgue Measurability of Functions on R	p. 83
Cantor Ternary Set and Cantor-Lebesgue Function	p. 85
Completion of Measure Space	p. 95
Complete Extension and Completion of a Measure Space	p. 95
Completion of the Borel Measure Space to the Lebesgue Measure Space	p. 98
Convergence a.e. and Convergence in Measure	p. 100
Convergence a.e.	p. 100
Almost Uniform Convergence	p. 104
Convergence in Measure	p. 107
Cauchy Sequences in Convergence in Measure	p. 112
Approximation by Step Functions and Continuous Functions	p. 115
The Lebesgue Integral	p. 127
Integration of Bounded Functions on Sets of Finite Measure	p. 127
Integration of Simple Functions	p. 127
Integration of Bounded Functions on Sets of Finite Measure	p. 131
Riemann Integrability	p. 140
Integration of Nonnegative Functions	p. 152
Lebesgue Integral of Nonnegative Functions	p. 152
Monotone Convergence Theorem	p. 154
Approximation of the Integral by Truncation	p. 162
Integration of Measurable Functions	p. 169
Lebesgue Integral of Measurable Functions	p. 169
Convergence Theorems	p. 178
Convergence Theorems under Convergence in Measure	p. 182
Approximation of the Integral by Truncation	p. 183
Translation and Linear Transformation in the Lebesgue Integral on R	p. 189
Integration by Image Measure	p. 193
Signed Measures	p. 202
Signed Measure Spaces	p. 202
Decomposition of Signed Measures	p. 208
Integration on a Signed Measure Space	p. 217
Absolute Continuity of a Measure	p. 224
The Radon-Nikodym Derivative	p. 224
Absolute Continuity of a Signed Measure Relative to a Positive Measure	p. 225
Properties of the Radon-Nikodym Derivative	p. 236
Differentiation and Integration	p. 245
Monotone Functions and Functions of Bounded Variation	p. 245
The Derivative	p. 245
Differentiability of Monotone Functions	p. 251
Functions of Bounded Variation	p. 261
Absolutely Continuous Functions	p. 270
Absolute Continuity	p. 270
Banach-Zarecki Criterion for Absolute Continuity	p. 273
Singular Functions	p. 276
Indefinite Integrals	p. 276
Calculation of the Lebesgue Integral by Means of the Derivative	p. 287
Length of Rectifiable Curves	p. 298
Convex Functions	p. 308
Continuity and Differentiability of a Convex Function	p. 308
Monotonicity and Absolute Continuity of a Convex Function	p. 317
Jensen's Inequality	p. 320
The Classical Banach Spaces	p. 323
Normed Linear Spaces	p. 323
Banach Spaces	p. 323
Banach Spaces on R[superscript k]	p. 326
The Space of Continuous Functions C([a, b])	p. 329
A Criterion for Completeness of a Normed Linear Space	p. 331
Hilbert Spaces	p. 333
Bounded Linear Mappings of Normed Linear Spaces	p. 334
Baire Category Theorem	p. 344
Uniform Boundedness Theorems	p. 347
Open Mapping Theorem	p. 350
Hahn-Banach Extension Theorems	p. 357
Semicontinuous Functions	p. 370
The L[superscript p] Spaces	p. 376
The L[superscript p] Spaces for p [set membership] (0, [infinity]	p. 376
The Linear Spaces L[superscript p] for p [set membership] [1, infinity]	p. 379
The L[superscript p] Spaces for p [set membership] [1, infinity])	p. 384
The Space L[superscript infinity]]	p. 393
The L[superscript p] Spaces for p [set membership] (0, 1)	p. 401
Extensions of Holder's Inequality	p. 406
Relation among the L[superscript p] Spaces	p. 412
The Modified L[superscript p] Norms for L[superscript p] Spaces with p [set membership] [1, infinity]	p. 412
Approximation by Continuous Functions	p. 414
L[superscript p] Spaces with p [set membership] (0, 1]	p. 417
The l[superscript p] Spaces	p. 422
Bounded Linear Functionals on the L[superscript p] Spaces	p. 429
Bounded Linear Functionals Arising from Integration	p. 429
Approximation by Simple Functions	p. 432
A Converse of Holder's Inequality	p. 434
Riesz Representation Theorem on the L[superscript p] Spaces	p. 437
Integration on Locally Compact Hausdorff Space	p. 445
Continuous Functions on a Locally Compact Hausdorff Space	p. 445
Borel and Radon Measures	p. 450
Positive Linear Functionals on C[subscript c](X)	p. 455
Approximation by Continuous Functions	p. 463
Signed Radon Measures	p. 467
The Dual Space of C(X)	p. 471
Extension of Additive Set Functions to Measures	p. 481
Extension of Additive Set Functions on an Algebra	p. 481
Additive Set Function on an Algebra	p. 481
Extension of an Additive Set Function on an Algebra to a Measure	p. 486
Regularity of an Outer Measure Derived from a Countably Additive Set Function on an Algebra	p. 486
Uniqueness of Extension of a Countably Additive Set Function on an Algebra to a Measure	p. 489
Approximation to a [sigma]-algebra Generated by an Algebra	p. 491
Outer Measure Based on a Measure	p. 494
Extension of Additive Set Functions on a Semialgebra	p. 496
Semialgebras of Sets	p. 496
Additive Set Function on a Semialgebra	p. 498
Outer Measures Based on Additive Set Functions on a Semialgebra	p. 502
Lebesgue-Stieltjes Measure Spaces	p. 505
Lebesgue-Stieltjes Outer Measures	p. 505
Regularity of the Lebesgue-Stieltjes Outer Measures	p. 509
Absolute Continuity and Singularity of a Lebesgue-Stieltjes Measure	p. 511
Decomposition of an Increasing Function	p. 519
Product Measure Spaces	p. 527
Existence and Uniqueness of Product Measure Spaces	p. 527
Integration on Product Measure Space	p. 531
Completion of Product Measure Space	p. 543
Convolution of Functions	p. 547
Some Related Theorems	p. 587
Measure and Integration on the Euclidean Space	p. 597
Lebesgue Measure Space on the Euclidean Space	p. 597
Lebesgue Outer Measure on the Euclidean Space	p. 597
Regularity Properties of Lebesgue Measure Space on R[superscript n]	p. 602
Approximation by Continuous Functions	p. 605
Lebesgue Measure Space on R[superscript n] as the Completion of a Product Measure Space	p. 609
Translation of the Lebesgue Integral on R[superscript n]	p. 610
Linear Transformation of the Lebesgue Integral on R[superscript n]	p. 612
Differentiation on the Euclidean Space	p. 620
The Lebesgue Differentiation Theorem on R[superscript n]	p. 620
Differentiation of Set Functions with Respect to the Lebesgue Measure	p. 632
Differentiation of the Indefinite Integral	p. 634
Density of Lebesgue Measurable Sets Relative to the Lebesgue Measure	p. 635
Signed Borel Measures on R[superscript n]	p. 641
Differentiation of Borel Measures with Respect to the Lebesgue Measure	p. 643
Change of Variable of Integration on the Euclidean Space	p. 649
Change of Variable of Integration by Differentiable Transformations	p. 649
Spherical Coordinates in R[superscript n]	p. 661
Integration by Image Measure on Spherical Surfaces	p. 667
Hausdorff Measures on the Euclidean Space	p. 675
Hausdorff Measures	p. 675
Hausdorff Measures on R[superscript n]	p. 675
Equivalent Definitions of Hausdorff Measure	p. 680
Regularity of Hausdorff Measure	p. 686
Hausdorff Dimension	p. 689
Transformations of Hausdorff Measures	p. 694
Hausdorff Measure of Transformed Sets	p. 694
1-dimensional Hausdorff Measure	p. 699
Hausdorff Measure of Jordan Curves	p. 700
Hausdorff Measures of Integral and Fractional Dimensions	p. 705
Hausdorff Measure of Integral Dimension and Lebesgue Measure	p. 705
Calculation of the n-dimensional Hausdorff Measure of a Unit Cube in R[superscript n]	p. 707
Transformation of Hausdorff Measure of Integral Dimension	p. 713
Hausdorff Measure of Fractional Dimension	p. 718
Bibliography	p. 727
Index	p. 729
Table of Contents provided by Ingram. All Rights Reserved.

Real Analysis

Theory Of Measure And Integration (2nd Edition)

At a Glance

Paperback

Industry Reviews

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How to return your order

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