Measure, Integration & Real Analysis, 2021a

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Chapter 12 Probability Measures 399 11 Give an example of a probability space (Ω, F, P) and X, Y ∈L 2 (P) such that σ 2 (X + Y) =σ 2 (X)+σ 2 (Y) but X and Y are not independent random variables. 12 Prove that if X and Y are random variables (possibly on two different probability spaces) and ˜X = Ỹ, then P X = P Y . 13 Suppose H : R → (0, 1) is a continuous one-to-one function satisfying conditions (a) through (d) of 12.29. Show that the function X : (0, 1) → R produced in the proof of 12.29 is the inverse function of H. 14 Suppose (Ω, F, P) is a probability space and X is a random variable. Prove that the following are equivalent: • ˜X is a continuous function on R. • ˜X is a uniformly continuous function on R. • P(X = t) =0 for every t ∈ R. • ( ˜X ◦ X)˜(s) =s for all s ∈ R. { 0 if x < 0, 15 Suppose α > 0 and h(x) = α 2 xe −αx if x ≥ 0. Let P = h dλ and let X be the random variable defined by X(x) =x for x ∈ R. (a) Verify that ∫ ∞ −∞ h dλ = 1. (b) Find a formula for the distribution function ˜X. (c) Find a formula (in terms of α) for EX. (d) Find a formula (in terms of α) for σ(X). 16 Suppose B is the σ-algebra of Borel subsets of [0, 1) and P is Lebesgue measure on ( [0, 1], B ) . Let {e k } k∈Z + be the family of functions defined by the fourth bullet point of Example 8.51 (notice that k = 0 is excluded). Show that the family {e k } k∈Z + is an i.i.d. 17 Suppose B is the σ-algebra of Borel subsets of (−π, π] and P is Lebesgue measure on ( (−π, π], B ) divided by 2π. Let {e k } k∈Z\{0} be the family of trigonometric functions defined by the third bullet point of Example 8.51 (notice that k = 0 is excluded). (a) Show that {e k } k∈Z\{0} is not an independent family of random variables. (b) Show that {e k } k∈Z\{0} is an identically distributed family. Measure, Integration & Real Analysis, by Sheldon Axler
Photo Credits • page vi: photos by Carrie Heeter and Sheldon Axler • page 1: digital sculpture created by Doris Fiebig; all rights reserved • page 13: photo by Petar Milošević; Creative Commons Attribution-Share Alike 4.0 International license; verified on 9 July 2019 at https:// commons.wikimedia.org/wiki/File:Mausoleum_of_Galla_Placidia_ceiling_mosaics.jpg • page 19: photo by Fagairolles 34; Creative Commons Attribution-Share Alike 4.0 International license; verified on 9 July 2019 at https:// commons.wikimedia.org/wiki/File:Saint-Affrique_pont.JPG • page 44: photo by Sheldon Axler • page 51: photo by James Mitchell; Creative Commons Attribution-Share Alike 2.0 Generic license; verified on 9 July 2019 at https:// commons.wikimedia.org/wiki/File:Beauvais_Cathedral_SE_exterior.jpg • page 68: photo by A. Savin; Creative Commons Attribution-Share Alike 3.0 Unported license; verified on 9 July 2019 at https:// commons.wikimedia.org/wiki/File:Moscow_05-2012_Mokhovaya_05.jpg • page 73: photo by Giovanni Dall’Orto; permission to use verified on 11 July 2019 at https://en.wikipedia.org/wiki/Maria_Gaetana_Agnesi • page 101: photo by Rafa Esteve; Creative Commons Attribution-Share Alike 4.0 International license; verified on 9 July 2019 at https:// commons.wikimedia.org/wiki/File:Trinity_College_-_Great_Court_02.jpg • page 102: photo by A. Savin; Creative Commons Attribution-Share Alike 3.0 Unported license; verified on 9 July 2019 at https:// commons.wikimedia.org/wiki/File:Spb_06-2012_University_Embankment_01.jpg • page 116: photo by Lucarelli; Creative Commons Attribution-Share Alike 3.0 Unported license; verified on 9 July 2019 at https:// commons.wikimedia.org/wiki/File:Palazzo_Carovana_Pisa.jpg • page 146: photo by Petar Milošević; Creative Commons Attribution-Share Alike 3.0 Unported license; verified on 9 July 2019 at https:// en.wikipedia.org/wiki/Lviv • page 152: photo by NonOmnisMoriar; Creative Commons Attribution-Share Alike 3.0 Unported license; verified on 9 July 2019 at https:// commons.wikimedia.org/wiki/File:Ecole_polytechnique_former_building_entrance.jpg Measure, Integration & Real Analysis, by Sheldon Axler 400
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Measure, Integration & Real Analysi
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About the Author Sheldon Axler was
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viii Contents 2D Lebesgue Measure 4
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x Contents 6E Consequences of Baire
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xii Contents 11 Fourier Analysis 33
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Preface for Instructors You are abo
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xvi Preface for Instructors • Cha
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Acknowledgments I owe a huge intell
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2 Chapter 1 Riemann Integration 1A
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4 Chapter 1 Riemann Integration m (
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6 Chapter 1 Riemann Integration whe
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8 Chapter 1 Riemann Integration 6 S
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10 Chapter 1 Riemann Integration Ca
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12 Chapter 1 Riemann Integration EX
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14 Chapter 2 Measures 2A Outer Meas
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16 Chapter 2 Measures The next resu
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18 Chapter 2 Measures Outer Measure
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20 Chapter 2 Measures Now we can pr
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≈ 7π Chapter 2 Measures Clearly
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24 Chapter 2 Measures 10 Prove that
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26 Chapter 2 Measures We have shown
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28 Chapter 2 Measures Borel Subsets
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30 Chapter 2 Measures Inverse image
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32 Chapter 2 Measures The definitio
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34 Chapter 2 Measures 2.42 Definiti
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38 Chapter 2 Measures EXERCISES 2B
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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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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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≈ 100π Chapter 10 Linear Maps on
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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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346 Chapter 11 Fourier Analysis 11.
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Page 395 and 396: Chapter 12 Probability Measures Pro
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Page 417 and 418: Bibliography The chapters of this b
Page 419 and 420: 404 Notation Index ∑ ∞ k=1 g k,
Page 421 and 422: Index Abel summation, 344 absolute
Page 423 and 424: 408 Index Hahn Decomposition Theore
Page 425 and 426: 410 Index random variable, 385 rang
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Measure, Integration & Real Analysis, 2021a

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