Multivariable Advanced Calculus

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206 THE ABSTRACT LEBESGUE INTEGRALcontinuous but ϕ ◦ g −1 is not measurable. (This is an example of measurable ◦continuous ≠ measurable.) Show there exist Lebesgue measurable sets which arenot Borel measurable. Hint: The function, ϕ is Lebesgue measurable. Now showthat Borel ◦ measurable = measurable.28. If A is m⌊S measurable, it does not follow that A is m measurable. Give anexample to show this is the case.29. If f is a nonnegative Lebesgue measurable function, show there exists g a Borelmeasurable function such that g (x) = f (x) a.e.
The Lebesgue Integral ForFunctions Of p Variables9.1 π SystemsThe approach to p dimensional Lebesgue measure will be based on a very elegant ideadue to Dynkin.Definition 9.1.1 Let Ω be a set and let K be a collection of subsets of Ω. ThenK is called a π system if ∅, Ω ∈ K and whenever A, B ∈ K, it follows A ∩ B ∈ K.For example, if R p = Ω, an example of a π system would be the set of all open sets.Another example would be sets of the form ∏ pk=1 A k where A k is a Lebesgue measurableset.The following is the fundamental lemma which shows these π systems are useful.Lemma 9.1.2 Let K be a π system of subsets of Ω, a set. Also let G be a collectionof subsets of Ω which satisfies the following three properties.1. K ⊆ G2. If A ∈ G, then A C ∈ G3. If {A i } ∞ i=1 is a sequence of disjoint sets from G then ∪∞ i=1 A i ∈ G.Then G ⊇ σ (K) , where σ (K) is the smallest σ algebra which contains K.Proof: First note that ifH ≡ {G : 1 - 3 all hold}then ∩H yields a collection of sets which also satisfies 1 - 3. Therefore, I will assume inthe argument that G is the smallest collection of sets satisfying 1 - 3, the intersection ofall such collections. Let A ∈ K and defineG A ≡ {B ∈ G : A ∩ B ∈ G} .I want to show G A satisfies 1 - 3 because then it must equal G since G is the smallestcollection of subsets of Ω which satisfies 1 - 3. This will give the conclusion that forA ∈ K and B ∈ G, A ∩ B ∈ G. This information will then be used to show that ifA, B ∈ G then A ∩ B ∈ G. From this it will follow very easily that G is a σ algebrawhich will imply it contains σ (K). Now here are the details of the argument.207
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4 CONTENTS5 Continuous Functions 85
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6 CONTENTS11.5 Integration Of Diffe
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8 CONTENTS
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10 INTRODUCTION
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12 SOME FUNDAMENTAL CONCEPTSthere i
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14 SOME FUNDAMENTAL CONCEPTSthat is
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16 SOME FUNDAMENTAL CONCEPTSFollow
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18 SOME FUNDAMENTAL CONCEPTSDefinit
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20 SOME FUNDAMENTAL CONCEPTSTheorem
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22 SOME FUNDAMENTAL CONCEPTSNext ob
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24 BASIC LINEAR ALGEBRA6 · (2, 6)(
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26 BASIC LINEAR ALGEBRA3.2 Subspace
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28 BASIC LINEAR ALGEBRATheorem 3.2.
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30 BASIC LINEAR ALGEBRAProof: This
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32 BASIC LINEAR ALGEBRAwhich showsL
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34 BASIC LINEAR ALGEBRAsuch thatl i
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36 BASIC LINEAR ALGEBRA3.4 Block Mu
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38 BASIC LINEAR ALGEBRAnumbers in (
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40 BASIC LINEAR ALGEBRAProof: Let (
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42 BASIC LINEAR ALGEBRABy Corollary
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44 BASIC LINEAR ALGEBRAProposition
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46 BASIC LINEAR ALGEBRADefinition 3
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48 BASIC LINEAR ALGEBRAcontrary to
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50 BASIC LINEAR ALGEBRALemma 3.6.2
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52 BASIC LINEAR ALGEBRA3. Let L ∈
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54 BASIC LINEAR ALGEBRATheorem 3.8.
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56 BASIC LINEAR ALGEBRALemma 3.8.9
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58 BASIC LINEAR ALGEBRA3.8.5 Orthon
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60 BASIC LINEAR ALGEBRAis in L (H 1
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62 BASIC LINEAR ALGEBRA3.8.7 Schur
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64 BASIC LINEAR ALGEBRAProof: Let {
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66 BASIC LINEAR ALGEBRANow for x, y
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68 BASIC LINEAR ALGEBRAby 3.47.Sinc
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70 BASIC LINEAR ALGEBRA8. A normal
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72 SEQUENCESTheorem 4.1.5 Suppose {
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74 SEQUENCESTheorem 4.1.8 Let {x n
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76 SEQUENCESTheorem 4.3.2 The inter
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78 SEQUENCESHowever, this sequence
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80 SEQUENCESGiven R is complete, th
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82 SEQUENCES10. Suppose A ⊆ R n a
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84 SEQUENCES
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86 CONTINUOUS FUNCTIONS≤ |a| |f (
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88 CONTINUOUS FUNCTIONSof f −1 (U
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90 CONTINUOUS FUNCTIONSProof: First
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92 CONTINUOUS FUNCTIONSlet C ∩ (
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94 CONTINUOUS FUNCTIONSProof: Suppo
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96 CONTINUOUS FUNCTIONSThen the fol
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98 CONTINUOUS FUNCTIONS5.6 Polynomi
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100 CONTINUOUS FUNCTIONSThus the ex
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102 CONTINUOUS FUNCTIONSTherefore,
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104 CONTINUOUS FUNCTIONSProof: Let
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106 CONTINUOUS FUNCTIONSand so ( )3
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108 CONTINUOUS FUNCTIONSLet( )a1b k
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110 CONTINUOUS FUNCTIONSTheorem 5.8
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112 CONTINUOUS FUNCTIONS5.9 Ascoli
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114 CONTINUOUS FUNCTIONSBy equicont
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116 CONTINUOUS FUNCTIONS11. By Theo
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118 CONTINUOUS FUNCTIONS28. Let X b
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120 CONTINUOUS FUNCTIONS
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122 THE DERIVATIVEor equivalently,|
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124 THE DERIVATIVEIt follows≡limt
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126 THE DERIVATIVEand soNow dividin
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128 THE DERIVATIVEprovided ||v|| is
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130 THE DERIVATIVEare both linear.T
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132 THE DERIVATIVE6.7.1 Some Standa
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134 THE DERIVATIVEDefinition 6.8.4
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136 THE DERIVATIVEAs implied above,
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138 THE DERIVATIVEwhich requires ea
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140 THE DERIVATIVETheorem 6.10.3 (i
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142 THE DERIVATIVEThen there exist
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144 THE DERIVATIVENow the idea is t
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146 THE DERIVATIVETheorem 6.11.5 If
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148 THE DERIVATIVEare linearly inde
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150 THE DERIVATIVEShow {x 1 , · ·
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152 THE DERIVATIVE25. Let (x, y) be
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154 MEASURES AND MEASURABLE FUNCTIO
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Page 182 and 183: 182 THE ABSTRACT LEBESGUE INTEGRALL
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256 THE LEBESGUE INTEGRAL FOR FUNCT
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258 BROUWER DEGREEDefinition 10.1.1
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260 BROUWER DEGREEIt is obvious tha
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262 BROUWER DEGREETherefore, for an
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264 BROUWER DEGREE= −A.Therefore,
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266 BROUWER DEGREEProof: From Lemma
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268 BROUWER DEGREEfor all t ∈ [0,
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270 BROUWER DEGREELemma 10.3.2 Let
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272 BROUWER DEGREELemma 10.4.2 Let
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274 BROUWER DEGREEwhich is clearly
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276 BROUWER DEGREEThe product formu
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278 BROUWER DEGREEProof: Let B (y,3
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280 BROUWER DEGREEand y /∈ C. Sin
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282 BROUWER DEGREEwhich is the same
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284 BROUWER DEGREEFor y ∈ S, let
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286 BROUWER DEGREELetdist ( )y, WjC
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288 BROUWER DEGREE∞∑∫∞∑
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290 BROUWER DEGREE11. Establish the
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292 BROUWER DEGREE19. Using Problem
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294 INTEGRATION OF DIFFERENTIAL FOR
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326 THE LAPLACE AND POISSON EQUATIO
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350 THE JORDAN CURVE THEOREMNow the
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352 THE JORDAN CURVE THEOREMA solid
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354 THE JORDAN CURVE THEOREMLemma 1
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356 THE JORDAN CURVE THEOREMwhere R
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358 THE JORDAN CURVE THEOREMa QHKbN
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360 THE JORDAN CURVE THEOREM
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362 LINE INTEGRALSLet x t ≡ tx 1
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364 LINE INTEGRALSActually, people
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366 LINE INTEGRALSand if t is in th
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368 LINE INTEGRALSProof: The functi
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370 LINE INTEGRALSProof of the clai
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372 LINE INTEGRALS(γ s − 2h + t
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374 LINE INTEGRALS≤≤n∑∫ tj|
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376 LINE INTEGRALSand so|F (γ (t))
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378 LINE INTEGRALSLet m be the last
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380 LINE INTEGRALSLemma 14.3.5 In t
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382 LINE INTEGRALSLemma 14.3.6 Let
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384 LINE INTEGRALSwheref (x, y) ≡
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386 LINE INTEGRALSwhere B δ is the
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388 LINE INTEGRALSto Green’s theo
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390 LINE INTEGRALSDefinition 14.4.2
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392 LINE INTEGRALS14.5 Interpretati
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394 LINE INTEGRALS✻a × b✣θc
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396 LINE INTEGRALSSuppose then that
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398 LINE INTEGRALSProposition 14.6.
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400 LINE INTEGRALSCombining these l
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402 LINE INTEGRALSIn this case the
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404 LINE INTEGRALSThen as in the pr
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406 LINE INTEGRALSConsider only h
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408 LINE INTEGRALSwhere |g (w)| <
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410 LINE INTEGRALS6. In the situati
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412 LINE INTEGRALS18. Using Problem
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414 LINE INTEGRALS24. Let f, g be a
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416 LINE INTEGRALSShow this equals
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418 LINE INTEGRALS
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420 HAUSDORFF MEASURES AND AREA FOR
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460 BIBLIOGRAPHY[21] Gurtin M. An i
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462 INDEXderivative, 307exact, 322i
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464 INDEXvectors, 25Vitaliconvergen
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Multivariable Advanced Calculus

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