Real Analysis: Theory of Measure and Integration (2nd Edition) Book

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