Verfügbarkeit: Real analysis

Real analysis: theory of measure and integration

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Bibliographische Detailangaben
1. Verfasser:	Yeh, James (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Singapore [u.a.] World Scientific 2014
Ausgabe:	3. ed.
Schlagworte:	Reelle Analysis Maßtheorie Integrationstheorie Lehrbuch
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	Erg. u.d.T.: Yeh, James: Problems and proofs in real analysis. - 1. Aufl. u.d.T.: Yeh, James: Lectures on real analysis
Beschreibung:	XXIII, 815 S.
ISBN:	9789814578547 9789814578530

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adam_text	Titel: Real analysis Autor: Yeh, James Jahr: 2014 Contents Preface to the First Edition xiii Preface to the Second Edition xvii Preface to the Third Edition xix List of Notations xxi 1 Measure Spaces 1 §0 Introduction......................................................................1 § 1 Measure on a ít-algebra of Sets..................................................3 [I] er-algebra of Sets..............................................................3 [II] Limits of Sequences of Sets..................................................4 [HI] Generation of cr-algebras....................................................6 [IV] Borei a-algebras............................................................9 [V] Measure on a cr-algebra......................................................11 [VI] Measures of a Sequence of Sets . .........................................14 [VÜ] Measurable Space and Measure Space....................................17 [Vili] Measurable Mapping......................................................19 [TX] Induction of Measure by Measurable Mapping...............22 §2 Outer Measures.................................29 [I] Construction of Measure by Means of Outer Measure...........29 [H] Regular Outer Measures ........................33 [HI] Metric Outer Measures ..........................35 [TV] Construction of Outer Measures......................38 §3 Lebesgue Measure on R..............................42 [I] Lebesgue Outer Measure on R........................42 [II] Some Properties of the Lebesgue Measure Space.............47 [III] Existence of Non-Lebesgue Measurable Sets...............51 [IV] Regularity of Lebesgue Outer Measure..................53 [V] Lebesgue Inner Measure on R.......................60 §4 Measurable Functions .............................72 [I] Measurability of Functions .........................72 [H] Operations with Measurable Functions ..................76 [III] Equality Almost Everywhere .......................81 [TV] Sequence of Measurable Functions....................82 vii v¡¡¡ Contents [V] Continuity and Borei and Lebesgue Measurability of Functions on R . . . 86 [VI] Cantor Ternary Set and Cantor-Lebesgue Function............ 88 §5 Completion of Measure Space .........................99 [I] Complete Extension and Completion of a Measure Space.........99 [II] Completion of the Borei Measure Space to the Lebesgue Measure Space 102 §6 Convergence a.e. and Convergence in Measure ................104 [I] Convergence a.e................................104 [II] Almost Uniform Convergence.......................108 [IE] Convergence in Measure .........................HI [IV] Cauchy Sequences in Convergence in Measure..............116 [V] Approximation by Step Functions and Continuous Functions.......119 2 The Lebesgue Integral 131 §7 Integration of Bounded Functions on Sets of Finite Measure..........131 [I] Integration of Simple Functions.......................131 [II] Integration of Bounded Functions on Sets of Finite Measure.......136 [ED] Riemann Integrability...........................145 §8 Integration of Nonnegative Functions......................159 [I] Lebesgue Integral of Nonnegative Functions................159 [II] Monotone Convergence Theorem .....................161 [III] Approximation of the Integral by Truncation...............169 §9 Integration of Measurable Functions......................177 [I] Lebesgue Integral of Measurable Functions.................177 [H] Convergence Theorems...........................186 [ID] Convergence Theorems under Convergence in Measure .........190 [IV] Approximation of the Integral by Truncation...............191 [V] Translation and Linear Transformation of the Lebesgue Integral on R . . 196 [VI] Integration by Image Measure.......................201 §10 Signed Measures ................................212 [I] Signed Measure Spaces...........................212 [II] Decomposition of Signed Measures ....................218 [IB] Integration on a Signed Measure Space..................227 §11 Absolute Continuity of a Measure .......................235 [I] The Radon-Nikodym Derivative ......................235 [II] Absolute Continuity of a Signed Measure Relative to a Positive Measure 236 [III] Properties of the Radon-Nikodym Derivative...............247 3 Differentiation and Integration 257 §12 Monotone Functions and Functions of Bounded Variation...........257 [I] The Derivative................................257 [II] Differentiability of Monotone Functions..................263 [III] Functions of Bounded Variation......................274 §13 Absolutely Continuous Functions.......................283 [I] Absolute Continuity.............................283 [II] Banach-Zarecki Criterion for Absolute Continuity ............286 Contents ix [III] Singular Functions.............................289 [TV] Indefinite Integrals.............................289 [V] Calculation of the Lebesgue Integral by Means of the Derivative.....300 [VI] Length of Rectifiable Curves .......................311 §14 Convex Functions ...............................323 [I] Continuity and Differentiability of a Convex Function...........323 [II] Monotonicity and Absolute Continuity of a Convex Function.......332 [OI] Jensen s Inequality.............................335 4 The Classical Banach Spaces 339 §15 Normed Linear Spaces.............................339 [I] Banach Spaces................................339 [H] Banach Spaces on R*............................342 [HI] The Space of Continuous Functions C([a, ¿]) ..............345 [TV] A Criterion for Completeness of a Normed Linear Space ........347 [V] Hilbert Spaces ...............................349 [VI] Bounded Linear Mappings of Normed Linear Spaces...........350 [VII] Baire Category Theorem.........................360 [Vili] Uniform Boundedness Theorems....................363 [TX] Open Mapping Theorem..........................366 [X] Hahn-Banach Extension Theorems.....................373 [XI] Semicontinuous Functions.........................386 §16 The LP Spaces .................................392 [I] The C Spaces for p e (0, oo) .......................392 [H] The Linear Spaces Cp for p 6 [1, oo)...................395 [HI] The Lp Spaces for p e [1, oo) ......................400 [IV] The Space L°°...............................410 [V] The Lp Spaces for p e (0,1)........................417 [VI] Extensions of Holder s Inequality.....................422 §17 Relation among the Lp Spaces.........................429 [I] The Modified Lp Norms for Lp Spaces with p e [1, oo] .........429 [0] Approximation by Continuous Functions .................431 [III] Lp Spaces with p e (0,1].........................435 [IV] The tp Spaces...............................439 §18 Bounded Linear Functional on the Lp Spaces ................448 [1] Bounded Linear Functionals Arising from Integration...........448 [H] Approximation by Simple Functions....................451 [IH] A Converse of Holder s Inequality.....................453 [TV] Riesz Representation Theorem on the Lp Spaces.............457 § 19 Integration on Locally Compact Hausdorff Space...............465 [I] Continuous Functions on a Locally Compact Hausdorff Space ......465 [H] Borei and Radon Measures.........................470 [ID] Positive Linear Functionals on Cc(X)...................475 [IV] Approximation by Continuous Functions.................483 [V] Signed Radon Measures..........................487 x Contents [VI] The Dual Space of C(X) .........................491 5 Extension of Additive Set Functions to Measures 501 §20 Extension of Additive Set Functions on an Algebra..............501 [I] Additive Set Function on an Algebra....................501 [II] Extension of an Additive Set Function on an Algebra to a Measure .... 506 [III] Regularity of an Outer Measure Derived from a Countably Additive Set Function on an Algebra ..........................506 [IV] Uniqueness of Extension of a Countably Additive Set Function on an Algebra to a Measure..........................509 [V] Approximation to a ct-algebra Generated by an Algebra .........511 [VI] Outer Measure Based on a Measure....................514 §21 Extension of Additive Set Functions on a Semialgebra ............516 [I] Semialgebras of Sets ............................516 [II] Additive Set Function on a Semialgebra..................518 [III] Outer Measures Based on Additive Set Functions on a Semialgebra . . . 522 §22 Lebesgue-Stieltjes Measure Spaces ......................525 [I] Lebesgue-Stieltjes Outer Measures.....................525 [II] Regularity of the Lebesgue-Stieltjes Outer Measures ...........529 [III] Absolute Continuity and Singularity of a Lebesgue-Stieltjes Measure . . 531 [IV] Decomposition of an Increasing Function.................539 §23 Product Measure Spaces............................548 [I] Existence and Uniqueness of Product Measure Spaces...........548 [II] Integration on Product Measure Space...................552 [III] Completion of Product Measure Space..................564 [IV] Convolution of Functions.........................568 [V] Some Related Theorems..........................608 6 Measure and Integration on the Euclidean Space 619 §24 Lebesgue Measure Space on the Euclidean Space...............619 [I] Lebesgue Outer Measure on the Euclidean Space .............619 [II] Regularity Properties of Lebesgue Measure Space on R .........624 [III] Approximation by Continuous Functions.................627 [IV] Lebesgue Measure Space on R as the Completion of a Product Measure Space...............................631 [V] Translation of the Lebesgue Integral on IR ................632 [VI] Linear Transformation of the Lebesgue Integral on R ..........634 §25 Differentiation on the Euclidean Space.....................643 [I] The Lebesgue Differentiation Theorem on R ...............643 [II] Differentiation of Set Functions with Respect to the Lebesgue Measure . 655 [HI] Differentiation of the Indefinite Integral..................657 [IV] Density of Lebesgue Measurable Sets Relative to the Lebesgue Measure 658 [V] Signed Borei Measures on R .......................664 [VI] Differentiation of Borei Measures with Respect to the Lebesgue Measure 666 §26 Change of Variable of Integration on the Euclidean Space...........673 Contents xi [I] Change of Variable of Integration by Differentiate Transformations . . . 673 [II] Spherical Coordinates in R ........................685 [HI] Integration by Image Measure on Spherical Surfaces...........691 7 Hausdorff Measures on the Euclidean Space 699 §27 Hausdorff Measures ..............................699 [I] Hausdorff Measures onR ..........................699 [II] Equivalent Definitions of Hausdorff Measure ...............704 [10] Regularity of Hausdorff Measure.....................710 [IV] Hausdorff Dimension...........................713 §28 Transformations of Hausdorff Measures....................718 [I] Hausdorff Measure of Transformed Sets ..................718 [11] 1-dimensional Hausdorff Measure.....................723 [III] Hausdorff Measure of Jordan Curves...................724 §29 Hausdorff Measures of Integral and Fractional Dimensions..........729 [I] Hausdorff Measure of Integral Dimension and Lebesgue Measure.....729 [II] Calculation of the n-dimensional Hausdorff Measure of a Unit Cube in R™ 731 [III] Transformation of Hausdorff Measure of Integral Dimension.......737 [IV] Hausdorff Measure of Fractional Dimension...............742 A Digital Expansions of Real Numbers 751 [I] Existence of p-digital Expansion......................751 [II] Uniqueness Question in /»-digital Representation.............754 [III] Cardinality of the Cantor Ternary Set...................757 B Measurability of Limits and Derivatives 761 [I] Borei Measurability of Limits of a Function ................761 [H] Borei Measurability of the Derivative of a Function............765 C Lipschitz Condition and Bounded Derivative 769 D Uniform Integrability 771 [I] Uniform Integrability ............................771 [II] Equi-integrability..............................777 [IH] Uniform Integrability on Finite Measure Spaces .............780 E Product-measurability and Factor-measurability 789 [I] Product-measurability and Factor-measurability of a Set..........789 [II] Product-measurability and Factor-measurability of a Function ......791 F Functions of Bounded Oscillation 793 [I] Partition of Closed Boxes in R .......................793 [O] Bounded Oscillation in R .........................795 [III] Bounded Oscillation on Subsets......................796 [IV] Bounded Oscillation on 1-dimensional Closed Boxes ..........797 [V] Bounded Oscillation and Measurability..................798 [VI] Evaluation of the Total Variation of an Absolutely Continuous Function . 799 xii Bibliography Index Contents 803 805
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spellingShingle	Yeh, James Real analysis theory of measure and integration Reelle Analysis (DE-588)4627581-2 gnd Maßtheorie (DE-588)4074626-4 gnd Integrationstheorie (DE-588)4138369-2 gnd
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title_full	Real analysis theory of measure and integration J. Yeh
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