(EBOOK PDF)Real Analysis 3rd Edition by Halsey Royden 0756638992 9780756638993 full chapters

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• ISBN 10:0756638992 

• ISBN 13:9780756638993 

• Author:Halsey Royden 

 This is the classic introductory graduate text. Heart of the book is measure theory and Lebesque integration.

Real Analysis 3rd Table of contents:

1 Measure Spaces
§0 Introduction ℝ
§1 Measure on a σ-algebra of Sets[I] σ-algebra of Sets[II] Limits of Sequences of Sets[III] Generation of σ-algebras[IV] Borel σ-algebras[V] Measure on a σ-algcbra[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 ℝ
§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
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 ℝ
§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 ℝ[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 ℝ
§12 Monotone Functions and Functions of Bounded Variation ℝ[I] The Derivative[II] Differentiability of Monotone Functions[III] Functions of Bounded Variation ℝ
§13 Absolutely Continuous Functions[I] Absolute Continuity[II] Banach-Zarecki Criterion for Absolute Continuity[III] Singular Functions[IV] Indefinite Integrals[V] Calculation of the Lebesgoe Integral by Means of the Derivative[VI] Length of Rectifiable Curves
§14 Convex Functions[I] Continuity and Differentiability of a Convex Function ℝ[II] Monotonicity and Absolute Continuity of a Convex Function ℝ[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[VII] Baire Category Theorem[VIII] Uniform Boundedness Theorems[IX] Open Mapping Theorem[X] Hahn-Battach Extension Theorems[XI] Semicontinuous Functions
§16 The Lp Spaces[I] The Lp Spaces for p ∈ (0, ∞)[II] The Linear Spaces Lp 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 ℓp Spaces
§18 Bounded linear Functionals on the Lp Spaces[I] Bounded Linear Functionals Arising from Integration ℝ[II] Approximation by Simple Functions[III] A Converse of Hölder’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 σ-algebra Generated by an Algebra[VI] Outer Measure Based on a Measure
§21 Extension of Additive Set Functions os 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[IV] Decomposition of an Increasing Function ℝ
§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[V] Some Related Theorems
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 ℝ[III] Approximation by Continuous Functions[IV] Lebesgue Measure Space on ℝ as the Completion of a Product Measure Space[V] Translation of the Lebesgue Integral on ℝ[VI] Linear Transformation of the Lebesgue Integral on ℝ
§25 Differentiation on the Euclidean Space[I] The Lebesgue Differentiation Theorem on ℝ[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 ℝ[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 ℝ[III] Integration by Image Measure on Spherical Surfaces
7 Hausdorff Measures on the Euclidean Space
§27 Hausdorff Measures[I] Hausdorff Measures on ℝ[II] Equivalent Definitions of Hausdorff Measure[III] Regularity of Hausdorff Measure[IV] Hausdorff Dimension ℝ
§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 ℝ[III] Transformation of Hausdorff Measure of Integral Dimension ℝ[IV] Hausdorff Measure of Fractional Dimension ℝ
A Digital Expansions of Real Numbers[I] Existence of p-digital Expansion ℝ[II] Uniqueness Question in p-digital Representation ℝ[III] Cardinality of the Cantor Ternary Set
B Measurability of Limits and Derivatives[I] Borel Measurability of Limits of a Function ℝ[II] Borel Measurability of the Derivative of a Function ℝ
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 ℝ
F Functions of Bounded Oscillation ℝ[I] Partition of Closed Boxes in ℝ[II] Bounded Oscillation in ℝ[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 ℝ
Bibliography
Index
 

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