Measure, Integration & Real Analysis, 2021a

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Contents About the Author vi Preface for Students Preface for Instructors xiii xiv Acknowledgments xviii 1 Riemann Integration 1 1A Review: Riemann Integral 2 Exercises 1A 7 1B Riemann Integral Is Not Good Enough 9 2 Measures 13 Exercises 1B 12 2A Outer Measure on R 14 Motivation and Definition of Outer Measure 14 Good Properties of Outer Measure 15 Outer Measure of Closed Bounded Interval 18 Outer Measure is Not Additive 21 Exercises 2A 23 2B Measurable Spaces and Functions 25 σ-Algebras 26 Borel Subsets of R 28 Inverse Images 29 Measurable Functions 31 Exercises 2B 38 2C Measures and Their Properties 41 Definition and Examples of Measures 41 Properties of Measures 42 Exercises 2C 45 Measure, Integration & Real Analysis, by Sheldon Axler vii
viii Contents 2D Lebesgue Measure 47 Additivity of Outer Measure on Borel Sets 47 Lebesgue Measurable Sets 52 Cantor Set and Cantor Function 55 Exercises 2D 60 2E Convergence of Measurable Functions 62 3 Integration 73 Pointwise and Uniform Convergence 62 Egorov’s Theorem 63 Approximation by Simple Functions 65 Luzin’s Theorem 66 Lebesgue Measurable Functions 69 Exercises 2E 71 3A Integration with Respect to a Measure 74 Integration of Nonnegative Functions 74 Monotone Convergence Theorem 77 Integration of Real-Valued Functions 81 Exercises 3A 84 3B Limits of Integrals & Integrals of Limits 88 4 Differentiation 101 Bounded Convergence Theorem 88 Sets of Measure 0 in Integration Theorems 89 Dominated Convergence Theorem 90 Riemann Integrals and Lebesgue Integrals 93 Approximation by Nice Functions 95 Exercises 3B 99 4A Hardy–Littlewood Maximal Function 102 Markov’s Inequality 102 Vitali Covering Lemma 103 Hardy–Littlewood Maximal Inequality 104 Exercises 4A 106 4B Derivatives of Integrals 108 Lebesgue Differentiation Theorem 108 Derivatives 110 Density 112 Exercises 4B 115 Measure, Integration & Real Analysis, by Sheldon Axler
Page 1 and 2: Measure, Integration & Real Analysi
Page 3: About the Author Sheldon Axler was
Page 7 and 8: x Contents 6E Consequences of Baire
Page 9 and 10: xii Contents 11 Fourier Analysis 33
Page 11 and 12: Preface for Instructors You are abo
Page 13 and 14: xvi Preface for Instructors • Cha
Page 15 and 16: Acknowledgments I owe a huge intell
Page 17 and 18: 2 Chapter 1 Riemann Integration 1A
Page 19 and 20: 4 Chapter 1 Riemann Integration m (
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Page 23 and 24: 8 Chapter 1 Riemann Integration 6 S
Page 25 and 26: 10 Chapter 1 Riemann Integration Ca
Page 27 and 28: 12 Chapter 1 Riemann Integration EX
Page 29 and 30: 14 Chapter 2 Measures 2A Outer Meas
Page 31 and 32: 16 Chapter 2 Measures The next resu
Page 33 and 34: 18 Chapter 2 Measures Outer Measure
Page 35 and 36: 20 Chapter 2 Measures Now we can pr
Page 37 and 38: ≈ 7π Chapter 2 Measures Clearly
Page 39 and 40: 24 Chapter 2 Measures 10 Prove that
Page 41 and 42: 26 Chapter 2 Measures We have shown
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Page 45 and 46: 30 Chapter 2 Measures Inverse image
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40 Chapter 2 Measures 23 Suppose f
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42 Chapter 2 Measures • Suppose X
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44 Chapter 2 Measures For convenien
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46 Chapter 2 Measures 5 Suppose (X,
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48 Chapter 2 Measures Thus |A ∪ G
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50 Chapter 2 Measures where the equ
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52 Chapter 2 Measures Lebesgue Meas
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54 Chapter 2 Measures In practice,
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56 Chapter 2 Measures 2.74 Definiti
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58 Chapter 2 Measures Now we can de
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60 Chapter 2 Measures EXERCISES 2D
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62 Chapter 2 Measures 2E Convergenc
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64 Chapter 2 Measures Proof Suppose
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66 Chapter 2 Measures Luzin’s The
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68 Chapter 2 Measures For each inte
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70 Chapter 2 Measures The next resu
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72 Chapter 2 Measures 9 Suppose F 1
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74 Chapter 3 Integration 3A Integra
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76 Chapter 3 Integration 3.6 Exampl
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78 Chapter 3 Integration 3.11 Monot
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80 Chapter 3 Integration Now we can
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82 Chapter 3 Integration The condit
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84 Chapter 3 Integration The inequa
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86 Chapter 3 Integration 12 Show th
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88 Chapter 3 Integration 3B Limits
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90 Chapter 3 Integration 3.27 Defin
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92 Chapter 3 Integration Suppose (X
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94 Chapter 3 Integration Proof Supp
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96 Chapter 3 Integration 3.42 Examp
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98 Chapter 3 Integration in other w
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100 Chapter 3 Integration 7 Let λ
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102 Chapter 4 Differentiation 4A Ha
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104 Chapter 4 Differentiation Suppo
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106 Chapter 4 Differentiation EXERC
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108 Chapter 4 Differentiation 4B De
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110 Chapter 4 Differentiation Deriv
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112 Chapter 4 Differentiation The n
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114 Chapter 4 Differentiation Proof
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Chapter 5 Product Measures Lebesgue
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118 Chapter 5 Product Measures 5.3
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120 Chapter 5 Product Measures Mono
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122 Chapter 5 Product Measures Now
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124 Chapter 5 Product Measures The
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126 Chapter 5 Product Measures 5.23
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128 Chapter 5 Product Measures EXER
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132 Chapter 5 Product Measures As y
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136 Chapter 5 Product Measures 5C L
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140 Chapter 5 Product Measures Volu
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142 Chapter 5 Product Measures This
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144 Chapter 5 Product Measures EXER
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Chapter 6 Banach Spaces We begin th
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148 Chapter 6 Banach Spaces The mat
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150 Chapter 6 Banach Spaces The def
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152 Chapter 6 Banach Spaces Entranc
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154 Chapter 6 Banach Spaces 14 Supp
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156 Chapter 6 Banach Spaces For b a
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158 Chapter 6 Banach Spaces We now
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160 Chapter 6 Banach Spaces 6.28 Ex
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162 Chapter 6 Banach Spaces EXERCIS
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164 Chapter 6 Banach Spaces Sometim
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166 Chapter 6 Banach Spaces 6.40 De
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168 Chapter 6 Banach Spaces 6.45 Ex
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170 Chapter 6 Banach Spaces EXERCIS
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172 Chapter 6 Banach Spaces 6D Line
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174 Chapter 6 Banach Spaces Discont
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176 Chapter 6 Banach Spaces The not
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178 Chapter 6 Banach Spaces If V is
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180 Chapter 6 Banach Spaces Then A
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182 Chapter 6 Banach Spaces 2 Suppo
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184 Chapter 6 Banach Spaces 6E Cons
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186 Chapter 6 Banach Spaces Because
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188 Chapter 6 Banach Spaces The nex
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190 Chapter 6 Banach Spaces 6.86 Pr
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192 Chapter 6 Banach Spaces A linea
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194 Chapter 7 L p Spaces 7A L p (μ
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196 Chapter 7 L p Spaces What we ca
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198 Chapter 7 L p Spaces 7.11 Examp
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200 Chapter 7 L p Spaces 6 Suppose
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202 Chapter 7 L p Spaces 7B L p (μ
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204 Chapter 7 L p Spaces L p (μ) I
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206 Chapter 7 L p Spaces Duality Re
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208 Chapter 7 L p Spaces EXERCISES
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210 Chapter 7 L p Spaces 18 Suppose
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212 Chapter 8 Hilbert Spaces 8A Inn
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214 Chapter 8 Hilbert Spaces As we
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216 Chapter 8 Hilbert Spaces The ne
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218 Chapter 8 Hilbert Spaces The or
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220 Chapter 8 Hilbert Spaces The pr
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222 Chapter 8 Hilbert Spaces 9 The
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224 Chapter 8 Hilbert Spaces 8B Ort
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226 Chapter 8 Hilbert Spaces In the
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228 Chapter 8 Hilbert Spaces 8.37 o
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230 Chapter 8 Hilbert Spaces The re
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232 Chapter 8 Hilbert Spaces 8.45 r
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234 Chapter 8 Hilbert Spaces Suppos
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236 Chapter 8 Hilbert Spaces 16 Sup
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238 Chapter 8 Hilbert Spaces • Fo
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240 Chapter 8 Hilbert Spaces • Su
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242 Chapter 8 Hilbert Spaces Proof
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244 Chapter 8 Hilbert Spaces 8.61 D
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246 Chapter 8 Hilbert Spaces A mome
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248 Chapter 8 Hilbert Spaces 8.73 E
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250 Chapter 8 Hilbert Spaces Riesz
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252 Chapter 8 Hilbert Spaces 10 (a)
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254 Chapter 8 Hilbert Spaces 23 Pro
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256 Chapter 9 Real and Complex Meas
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≈ 100e Chapter 9 Real and Complex
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Chapter 10 Linear Maps on Hilbert S
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282 Chapter 10 Linear Maps on Hilbe
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340 Chapter 11 Fourier Analysis 11A
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342 Chapter 11 Fourier Analysis Hil
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344 Chapter 11 Fourier Analysis 11.
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348 Chapter 11 Fourier Analysis Sol
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350 Chapter 11 Fourier Analysis Fou
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352 Chapter 11 Fourier Analysis In
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354 Chapter 11 Fourier Analysis 12
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356 Chapter 11 Fourier Analysis Now
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362 Chapter 11 Fourier Analysis 11
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364 Chapter 11 Fourier Analysis (b)
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366 Chapter 11 Fourier Analysis Not
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368 Chapter 11 Fourier Analysis Con
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370 Chapter 11 Fourier Analysis Our
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372 Chapter 11 Fourier Analysis The
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374 Chapter 11 Fourier Analysis Fou
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376 Chapter 11 Fourier Analysis whe
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378 Chapter 11 Fourier Analysis 3 S
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Chapter 12 Probability Measures Pro
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382 Chapter 12 Probability Measures
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Photo Credits • page vi: photos b
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Bibliography The chapters of this b
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404 Notation Index ∑ ∞ k=1 g k,
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Index Abel summation, 344 absolute
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408 Index Hahn Decomposition Theore
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410 Index random variable, 385 rang
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Measure, Integration & Real Analysis, 2021a

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