Real analysis: theory of measure and integration

Contents
Preface to the First Edition
Preface to the Second Edition
Preface to the Third Edition
List of Notations
1 Measure Spaces
	§0 Introduction
	§1 Measure on a  -algebra of Sets
		[I]  -algebra of Sets
		[ll] Limits of Sequences of Sets
		[III] Generation of  -algebras
		[IV] Borel   -algebras
		[V] Measure on a  -algebra
		[VI] Measures of a Sequence of Sets
		[VII] Measurable Space and Measure Space
		[VIII] Measurable Mapping
		[IX] Induction of Measure by Measurable Mapping
	§2 Outer Measures
		[I] Construction of Measure by Means of Outer Measure
		[II] Regular Outer Measures
		[III] Metric Outer Measures
		[IV] Construction of Outer Measures
	§3 Lebesgue Measure on R
		[I] Lebesgue Outer Measure on R
		[II] Some Properties of the Lebesgue Measure Space
		[III] Existence of Non-Lebesgue Measurable Sets
		[IV] Regularity of Lebesgue Outer Measure
		[V] Lebesgue Inner Measure on R
	§4 Measurable Functions
		[I] Measurability of Functions
		[II] Operations with Measurable Functions
		[III] Equality Almost Everywhere
		[IV] Sequence of Measurable Functions
		[V] Continuity and Borel and Lebesgue Measurability of Functions on R
		[VI] Cantor Ternary Set and Cantor-Lebesgue Function
			[VI.1] Construction of Cantor Ternary Set
			[VI.2] Cantor-Lebesgue Function
	§5 Completion of Measure Space
		[I] Complete Extension and Completion of a Measure Space
		[II] Completion of the Borel Measure Space to the Lebesgue Measure Space
	§6 Convergence a.e. and Convergence in Measure
		[I] Convergence a.e.
		[II] Almost Uniform Convergence
		[III] Convergence in Measure
		[IV] Cauchy Sequences in Convergence in Measure
		[V] Approximation by Step Functions and Continuous Functions
			[V.1] Approximation of Lebesgue Measurable Functions by Step Functions
			[V.2] Approximation of Lebesgue Measurable Functions by Continuous Functions
2 The Lebesgue Integral
	§7 Integration of Bounded Functions on Sets of Finite Measure
		[I] Integration of Simple Functions
		[II] Integration of Bounded Functions on Sets of Finite Measure
		[III] Riemann Integrability
	§8 Integration of Nonnegative Functions
		[I] Lebesgue Integral of Nonnegative Functions
		[II] Monotone Convergence Theorem
		[III] Approximation of the Integral by Truncation
	§9 Integration of Measurable Functions
		[I] Lebesgue Integral of Measurable Functions
		[II] Convergence Theorems
		[III] Convergence Theorems under Convergence in Measure
		[IV] Approximation of the Integral by Truncation
		[V] Translation and Linear Transformation of the Lebesgue Integral on R
		[VI] Integration by Image Measure
	§ 10 Signed Measures
		[I] Signed Measure Spaces
		[II] Decomposition of Signed Measures
		[III] Integration on a Signed Measure Space
	§ 11 Absolute Continuity of a Measure
		[I] The Radon-Nikodym Derivative
		[II] Absolute Continuity of a Signed Measure Relative to a Positive Measure
		[III] Properties of the Radon-Nikodym Derivative
3 Differentiation and Integration
	§ 12 Monotone Functions and Functions of Bounded Variation
		[I] The Derivative
		[II] Differentiability of Monotone Functions
		[III] Functions of Bounded Variation
	§ 13 Absolutely Continuous Functions
		[I] Absolute Continuity
		[II] Banach-Zarecki Criterion for Absolute Continuity
		[III] Singular Functions
		[IV] Indefinite Integrals
		[V] Calculation of the Lebesgue Integral by Means of the Derivative
			[V.1] The Fundamental Theorem of Calculus
			[V.2] Integration by Parts
			[V.3] Change of Variables
		[VI] Length of Rectifiable Curves
	§ 14 Convex Functions
		[I] Continuity and Differentiability of a Convex Function
		[II] Monotonicity and Absolute Continuity of a Convex Function
		[III] Jensen's Inequality
4 The Classical Banach Spaces
	§ 15 Normed Linear Spaces
		[I] Banach Spaces
		[II] Banach Spaces on R
		[III] The Space of Continuous Functions C([a, b])
		[IV] A Criterion for Completeness of a Normed Linear Space
		[V] Hilbert Spaces
		[VI] Bounded Linear Mappings of Normed Linear Spaces
			[V.1] Continuous Linear Mappings
			[V.2] Bounded Linear Mappings
			[VI.3] Equivalence of Continuity and Roundedness of a Linear Mapping
			[VI.4] Infimum of the Bounds of a Linear Mapping
			[VI.5] Normed Linear Space of Bounded Linear Mappings
			[VII.6] Dual of a Normed Linear Space
		[VII] Baire Category Theorem
		[VIII] Uniform Boundedness Theorems
		[IX] Open Mapping Theorem
		[X] Hahn-Banach Extension Theorems
			[X.1] Hahn-Banach Extension Theorem for Real Linear Spaces
			[X.2] Hahn-Banach Extension Theorem for Complex Linear Spaces
			[X.3] Bounded Linear Mappings of a Dual Space into a Dual Space
		[XI] Semicontinuous Functions
	§ 16 The LP Spaces
		[I] The  P Spaces for p   (0,  )
		[II] The Linear Spaces  P for p   [1,  )
		[III] The LP Spaces for p   [1,  )
		[IV] The Space L
		[V] The LP Spaces for p   (0, 1)
		[VI] Extensions of Holder's Inequality
	§ 17 Relation among the LP Spaces
		[I] The Modified LP Norms for LP Spaces with p   [1,  ]
		[II] Approximation by Continuous Functions
		[III] LP Spaces with p   (0, 1]
		[IV] The  P Spaces
	§ 18 Bounded Linear Functionals on the LP Spaces
		[I] Bounded Linear Functionals Arising from Integration
		[II] Approximation by Simple Functions
		[III] A Converse of Holder's Inequality
		[IV] Riesz Representation Theorem on the LP Spaces
	§ 19 Integration on Locally Compact Hausdorff Space
		[I] Continuous Functions on a Locally Compact Hausdorff Space
		[II] Borel and Radon Measures
		[III] Positive Linear Functionals on Cc(X)
		[IV] Approximation by Continuous Functions
		[V] Signed Radon Measures
		[VI] The Dual Space of C(X)
5 Extension of Additive Set Functions to Measures
	§20 Extension of Additive Set Functions on an Algebra
		[I] Additive Set Function on an Algebra
		[II] Extension of an Additive Set Function on an Algebra to a Measure
		[III] Regularity of an Outer Measure Derived from a Countably Additive Set Function on an Algebra
		[IV] Uniqueness of Extension of a Countably Additive Set Function on an Algebra to a Measure
		[V] Approximation to a -algebra Generated by an Algebra
		[VI] Outer Measure Based on a Measure
	§21 Extension of Additive Set Functions on a Semialgebra
		[I] Semialgebras of Sets
		[II] Additive Set Function on a Semialgebra
		[III] Outer Measures Based on Additive Set Functions on a Semialgebra
	§22 Lebesgue-Stieltjes Measure Spaces
		[I] Lebesgue-Stieltjes Outer Measures
		[II] Regularity of the Lebesgue-Stieltjes Outer Measures
		[III] Absolute Continuity and Singularity of a Lebesgue-Stieltjes Measure
			[III.1] Absolute Continuity of Lebesgue-Stieltjes Measures
			[III.2] Singularity of Lebesgue-Stieltjes Measures
		[IV] Decomposition of an Increasing Function
	§ 23 Product Measure Spaces
		[I] Existence and Uniqueness of Product Measure Spaces
		[II] Integration on Product Measure Space
		[III] Completion of Product Measure Space
		[IV] Convolution of Functions
			[IV.1] Convolution of Integrable Functions
			[IV.2] Convolution of LP-Functions
			[IV.3] Approximate Identity in Convolution Product
			[IV.4] Approximate Identity Relative to Pointwise Convergence
		[V] Some Related Theorems
			[V.1] Cavalieri's Formula and Extensions
			[V.2] Minkowski's Inequality for Integrals
			[V.3] Approximation by Product Simple Functions
6 Measure and Integration on the Euclidean Space
	§24 Lebesgue Measure Space on the Euclidean Space
		[I] Lebesgue Outer Measure on the Euclidean Space
		[II] Regularity Properties of Lebesgue Measure Space on Rn
		[III] Approximation by Continuous Functions
		[IV] Lebesgue Measure Space on Rn as the Completion of a Product Measure Space
		[V] Translation of the Lebesgue Integral on Rn
		[VI] Linear Transformation of the Lebesgue Integral on Rn
	§ 25 Differentiation on the Euclidean Space
		[I] The Lebesgue Differentiation Theorem on Rn
		[II] Differentiation of Set Functions with Respect to the Lebesgue Measure
		[III] Differentiation of the Indefinite Integral
		[IV] Density of Lebesgue Measurable Sets Relative to the Lebesgue Measure
		[V] Signed Borel Measures on Rn
		[VI] Differentiation of Borel Measures with Respect to the Lebesgue Measure
	§ 26 Change of Variable of Integration on the Euclidean Space
		[I] Change of Variable of Integration by Differentiable Transformations
		[II] Spherical Coordinates in Rn
		[III] Integration by Image Measure on Spherical Surfaces
7 Hausdorff Measures on the Euclidean Space
	§27 Hausdorff Measures
		[I] Hausdorff Measures on Rn
		[II] Equivalent Definitions of Hausdorff Measure
			[II.1] Covering by Closed Sets and by Open Sets
			[II.2] Covering by Convex Sets
		[III] Regularity of Hausdorff Measure
		[IV] Hausdorff Dimension
	§28 Transformations of Hausdorff Measures
		[I] Hausdorff Measure of Transformed Sets
		[II] 1-dimensional Hausdorff Measure
		[III] Hausdorff Measure of Jordan Curves
	§29 Hausdorff Measures of Integral and Fractional Dimensions
		[I] Hausdorff Measure of Integral Dimension and Lebesgue Measure
		[II] Calculation of the n-dimensional Hausdorff Measure of a Unit Cube in Rn
		[III] Transformation of Hausdorff Measure of Integral Dimension
		[IV] Hausdorff Measure of Fractional Dimension
A Digital Expansions of Real Numbers
	[I] Existence of p-digital Expansion
	[II] Uniqueness Question in p-digital Representation
	[III] Cardinality of the Cantor Ternary Set
B Measurability of Limits and Derivatives
	[I] Borel Measurability of Limits of a Function
	[ll] Borel Measurability of the Derivative of a Function
C Lipschitz Condition and Bounded Derivative
D Uniform Integrability
	[I] Uniform Integrability
	[II] Equi-integrability
	[III] Uniform Integrability on Finite Measure Spaces
E Product-measurability and Factor-measurability
	[I] Product-measurability and Factor-measurability of a Set
	[II] Product-measurability and Factor-measurability of a Function
F Functions of Bounded Oscillation
	[I] Partition of Closed Boxes in Rn
	[II] Bounded Oscillation in Rn
	[III] Bounded Oscillation on Subsets
	[IV] Bounded Oscillation on 1-dimensional Closed Boxes
	[V] Bounded Oscillation and Measurability
	[VI] Evaluation of the Total Variation of an Absolutely Continuous Function
Bibliography
Index