CONTENTS Bibilography for Chapter 3 End of Chapter Problems 4 Metric Spaces 4.1 Convergence 4.1.1Co nce of functions 4.2 Completeness 4.2.1 Completion of a metric space 4.3 Compactness 4.4 Connectedness 4.5 Normed Vector Spaces 4.5.1 Convex sets 4.5.2 A finite dimensional vector space: Rn 的70888299 4.5.3 Series 4.5.4 An infinite dimensional vector space: ep 4.6 Continuous Functions 105 4.6.1 Intermediate value theorem 108 4.6.2 Extreme value theorem 110 4.6.3 Uniform continuity 4.7 Hemicontinuous Correspondences 4.7.1 Theorem of the maximum 4.8 Fixed Points and Contraction Mappings 4.8.1 Fixed points of functions 4.8.2 Contractions 4.8.3 Fixed points of correspondences 4.9 Appendix- Proofs in Chapter 4 138 4.10 Bibilography for Chapter 4 144 4.11 End of Chapter Problems 145 5 Measure Spaces 14 5.1.1 Outer measure 5.1.2 -measurable sets 154 5. 1.3 Lebesgue meets borel 5.1.4 L-measurable mappings 159 5.2 Lebesgue Integration 170 5.2.1 Riemann integrals 170 5.2.2 Lebesgue integrals4 CONTENTS 3.6 Bibilography for Chapter 3 . . . . . . . . . . . . . . . . . . . . 63 3.7 End of Chapter Problems. . . . . . . . . . . . . . . . . . . . . 64 4 Metric Spaces 65 4.1 Convergence ..................... . . . . . . . 68 4.1.1 Convergence of functions . . . . . . . . . . . . . . . . . 75 4.2 Completeness .................... . . . . . . . 77 4.2.1 Completion of a metric space. . . . . . . . . . . . . . . 80 4.3 Compactness .................... . . . . . . . 82 4.4 Connectedness .................... . . . . . . . 87 4.5 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 88 4.5.1 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . 92 4.5.2 A finite dimensional vector space: Rn . . . . . . . . . . 93 4.5.3 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.5.4 An infinite dimensional vector space: !p . . . . . . . . . 99 4.6 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . 105 4.6.1 Intermediate value theorem . . . . . . . . . . . . . . . 108 4.6.2 Extreme value theorem . . . . . . . . . . . . . . . . . . 110 4.6.3 Uniform continuity . . . . . . . . . . . . . . . . . . . . 111 4.7 Hemicontinuous Correspondences . . . . . . . . . . . . . . . . 113 4.7.1 Theorem of the Maximum . . . . . . . . . . . . . . . . 122 4.8 Fixed Points and Contraction Mappings . . . . . . . . . . . . 127 4.8.1 Fixed points of functions . . . . . . . . . . . . . . . . . 127 4.8.2 Contractions . . . . . . . . . . . . . . . . . . . . . . . . 130 4.8.3 Fixed points of correspondences . . . . . . . . . . . . . 132 4.9 Appendix - Proofs in Chapter 4 . . . . . . . . . . . . . . . . . 138 4.10 Bibilography for Chapter 4 . . . . . . . . . . . . . . . . . . . . 144 4.11 End of Chapter Problems . . . . . . . . . . . . . . . . . . . . 145 5 Measure Spaces 149 5.1 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.1.1 Outer measure . . . . . . . . . . . . . . . . . . . . . . 151 5.1.2 L−measurable sets . . . . . . . . . . . . . . . . . . . . 154 5.1.3 Lebesgue meets borel . . . . . . . . . . . . . . . . . . . 158 5.1.4 L-measurable mappings . . . . . . . . . . . . . . . . . 159 5.2 Lebesgue Integration . . . . . . . . . . . . . . . . . . . . . . . 170 5.2.1 Riemann integrals . . . . . . . . . . . . . . . . . . . . . 170 5.2.2 Lebesgue integrals . . . . . . . . . . . . . . . . . . . . 172