Measure and Integration Summary Jacob Shapiro September 4, 2013 Abstract This is a (very rough) translation of Prof. Michael Struwe’s German lecture Notes into English, as preparation for the final test (held in August 2013). Struwe followed rather closely the construction presented in the book of Evans and Gariepy and also had some inspiration from Amann and Escher. Contents 1 σ -Algebras and Measures 10 1.1 The Abstract Space ......... 10 1.1.1 D Measure .......... 10 1.1.2 E ............... 10 1.1.3 R σ-Subadditivity ...... 11 1.1.4 R Monotonicity ....... 11 1.1.5 R ............... 11 1.1.6 D Measurability of a Set (Caratheodory) 11 1.1.7 R ............... 11 1.1.8 E ............... 11 1.1.9 R ............... 12 1.1.10 D Algebra .......... 12 1.1.11 R ............... 12 1.1.12 E ............... 12 1 CONTENTS CONTENTS 1.1.13 T ............... 13 1.1.14 D Measure Spaces ..... 15 1.1.15 T Properties of µ on Σ ... 15 1.1.15.1 σ-additivity .... 15 1.1.15.2 Continuity from in- side ......... 16 1.1.15.3 Continuity from out- side ......... 16 1.1.16 E ............... 17 1.1.17 D µ-Measure Zero Sets (Null Sets) .............. 17 1.1.18 T Sets of Measure Zero are Measurable .......... 17 1.2 Construction of Measures ...... 17 1.2.1 D Covering Classes ..... 17 1.2.2 E Different Covering Classes 17 1.2.2.1 Of Rn ........ 17 1.2.2.2 Trivial Covering Class 18 1.2.3 T Forming Measures from Covering Classes ....... 18 1.2.4 E Generating the Measure defined in 1 .......... 19 1.2.5 D Pre-measure ....... 19 1.2.5.1 σ-finite pre-measure 20 1.2.6 R ............... 20 1.2.7 T Carathéodory-Hahn Ex- tension Theorem ....... 20 1.2.8 T Uniqueness of Caratheodory Measure ............ 21 1.2.9 R ............... 23 2 CONTENTS CONTENTS 1.2.9.1 Overlapping Σ’s .. 23 1.2.9.2 Cannot Generalize Beyond A ..... 23 1.2.10 E to 1.2.9.2 ......... 23 1.3 The Lebesgue Measure ........ 23 1.3.1 D Elementary Volume Pre- measure ............ 23 1.3.1.1 n-dimensional Box (k-cell) ....... 23 1.3.1.2 The Volume of an n-dimensional Box 24 1.3.1.3 An elementary Set 24 1.3.2 R ............... 24 1.3.3 R ............... 24 1.3.4 D The Lebesgue Measure Ln 24 1.3.5 T Open Sets are Countable Unions of k-cells ....... 24 1.3.6 C Borel σ-Algebra ..... 25 1.3.7 D Borel Measures ...... 25 1.3.8 T Approximation from Out- side by Open Sets ...... 25 1.3.9 D Regular Borel Measure . 25 1.3.10 C The Lebesgue Measure is a regular Borel measure ... 26 1.3.11 T Another Characterization of Measurability ....... 26 1.3.12 C Another Characterization of Measurability ....... 26 1.3.13 C Another Characterization of Measurability ....... 26 3 CONTENTS CONTENTS 1.3.14 Comparison with the Jordan Measure ............ 26 1.3.14.1 D The Jordan Mea- sure ......... 26 1.3.14.2 T Comparison with Lebesgue-measure . 27 1.3.15 T Invariance for Rigid Trans- formations ........... 27 1.4 The Limits of Measurability .... 27 1.4.1 E Vitali Sets ......... 27 1.4.2 Banach-Tarski ........ 27 1.5 The Lebesgue-Stieltjes Measure .. 27 1.5.1 D Distance between two sets 27 1.5.2 D Metric Measure ..... 28 1.5.3 T Caratheodory Criterion for Borel Measures ........ 28 1.5.4 T The Lebesgue-Stieltjes Mea- sure is a regular Borel measure 29 1.5.5 T Lebesgue-Stieltjes measure on half-open intervals .... 30 1.5.6 E Familiar Examples .... 31 1.6 The Hausdorff Measure ....... 32 1.6.0.1 D The Hausdorff Measure ...... 32 1.6.1 T The Hausdorff Measure is a regular Borel measure on Rn .............. 32 1.6.1.1 Step 1 ........ 33 1.6.1.2 Step 2 ........ 33 1.6.1.3 Step 3 ........ 33 4 CONTENTS CONTENTS 1.6.2 T Preparation for Hausdroff Dimension ........... 34 1.6.3 E Compact Sets ....... 35 1.6.4 R Connection between Lebesgue and Hausdorff measures ... 35 1.6.5 E Different Dimension Haus- dorff Measure ......... 36 1.6.6 D Hausdorff Dimension .. 36 1.6.7 E The Cantor “Dust” .... 36 1.7 Radon Measures ........... 37 1.7.1 D The Radon Measure ... 37 1.7.2 E ............... 38 1.7.3 T Approximation of Radon Measures by Open or Com- pact Sets ........... 39 1.8 Topological Definitions ........ 40 1.8.1 D Locally Compact Space . 40 1.8.2 D Semicontinuous Functions 40 1.8.2.1 Lower Semincontin- uous ......... 40 1.8.2.2 Upper Semicontin- uous ......... 40 1.8.3 D Support .......... 40 1.8.4 D Continuous Functions with Compact Support ...... 40 2 Measurable Functions 40 2.1 Definition and Elementary Charac- teristics ................. 40 5 CONTENTS CONTENTS 2.1.1 T Approximation with Step Functions ........... 40 2.1.2 R Monotone Convergence and Real Functions ........ 41 2.2 Lusin’s Theorem and Egoroff’s The- orem .................. 41 2.2.1 Egorov’s (Egoroff) Theorem 42 2.2.2 Lusin’s Theorem ....... 43 2.3 Convergence in Measure ....... 44 2.3.1 D Convergence In Measure 44 2.3.2 T Convergence Almost Ev- erywhere Implies In Measure 44 2.3.3 T Convergence in Measure gives a subsequence that Con- verges Almost Everywhere . 45 3 Integration 45 3.1 Definition and Elementary Charac- teristics ................. 45 3.1.1 D σ-Step Function ..... 45 3.1.2 D Integral for σ-Step Func- tions .............. 45 3.2 Convergence Theorems ....... 46 3.3 Absolute continuity of the Integral . 46 3.4 Vitali’s Theorem ........... 46 3.4.1 D Uniformly Integrable .. 46 3.4.2 T Vitali’s Theorem ..... 46 3.5 Lp (Ω; µ) spaces ............ 47 4 Complex Measures (Rudin RCA Chapter 6) 47 6 CONTENTS CONTENTS 4.1 Total Variation ............ 47 4.1.1 D A Partition ........ 47 4.1.2 D A Complex Measure ... 48 4.1.3 D Total Variation Measure of µ ............... 48 4.1.4 T jµj is a positive measure on M .............. 48 4.1.5 T ............... 48 4.1.6 D ............... 48 4.1.7 Positive and Negative Variations– Jordan Decomposition of µ . 49 4.2 Absolute Continuity ......... 49 4.2.1 σ-finite Measures ....... 49 4.2.2 The Lebesgue-Radon-Nikodym Theorem ............ 49 4.2.3 Theorem ............ 50 4.2.4 Theorem ............ 50 4.2.5 Theorem ............ 50 4.2.6 The Hahn Decomposition The- orem .............. 50 5 Intergration on Product Measures (Rudin RCA Chapter 8) 51 5.1 Measurability on Cartesian Products 51 5.1.1 A Rectangle in X × Y .... 51 5.1.2 A Measurable Rectangle .. 51 5.1.3 The Class of all Elementary Sets .............. 51 5.1.4 S × T ............. 51 5.1.5 A monotone Class ...... 51 5.1.6 x-section and y-section ... 51 7 CONTENTS CONTENTS 5.1.7 Theorem ............ 52 5.1.8 Theorem ............ 52 5.1.9 Theorem ............ 52 5.2 Product Measures .......... 52 5.2.1 Theorem ............ 52 5.2.2 The Product of Measures .. 52 5.3 The Fubini Theorem ......... 53 5.3.1 The Fubini Theorem ..... 53 5.3.2 Counterexamples ....... 53 5.4 Completion of Product Measures . 53 5.4.1 Theorem ............ 53 5.5 Convolutions .............. 53 5.5.1 Theorem–Convolutions ... 53 6 Product Measures, Multiple Integrals 54 6.1 Fubini’s Theorem ........... 54 6.2 Convolution .............. 54 7 Differentiation (Rudin RCA) 54 7.1 Derivatives of Measures ....... 54 7.1.1 Lemma ............. 54 7.1.2 Theorem ............ 54 7.1.3 Weak L1 ............ 55 7.1.3.1 The Maximal Func- tion (Hardy-Littlewood) 55 7.1.4 Lebesgue Points ....... 55 7.1.4.1 Example ...... 55 7.1.5 Theorem–Lebesgue Point a.e. 55 7.1.6 Theorem–The Radon-Nikodym Derivative ........... 55 7.1.7 Nicely Shrinking Sets .... 56 8 CONTENTS CONTENTS 7.1.8 Theorem ............ 56 7.1.9 Theorem ............ 56 7.1.10 Metric Density ........ 56 7.1.11 Theorem ............ 57 7.1.12 Theorem ............ 57 7.1.13 Theorem ............ 57 7.1.14 Theorem–The Fundamental Theorem of Calculus ..... 57 7.1.14.1 Counter Examples 57 7.1.14.1.1 a ..... 57 7.1.14.1.2 b–Denjoy and Perron .. 57 7.1.15 Absolutely Continuous Func- tions .............. 58 7.1.16 Theorem ............ 58 7.1.17 Total Variation Function of a Function ........... 58 7.1.18 Theorem ............ 58 7.1.19 Theorem ............ 59 7.1.20 Theorem ............ 59 8 Differentiation of Measures 59 8.1 Differentiability of the Lebesgue In- tegral .................. 59 8.2 Differentiation of Radon Measures . 59 8.3 Differentiation of Absolutely Con- tinuous Measures on R ........ 59 8.4 Lebesgue Decomposition ....... 59 8.5 The Radon-Nikodym Theorem ... 59 9 1 σ -ALGEBRAS AND MEASURES Conventional Notation • The numbers in parenthesis after every heading are a reference to Struwe’s own number codes, which I wanted to retain for convenience. −1 • Cc (X) = f 2 C (X; C): closureX f (Cn f0g) is compact • D : definition • R : remark • P : proof • E : example • T : theorem • A : application • C : Corollary 1 σ -Algebras and Measures 1.1 The Abstract Space Let X be a set, and let 2X be the power set of X. 1.1.1 D Measure (strw def. 1.1.1) A map µ : 2X ! [0; 1] is a measure on X, in case the following conditions hold: 1.