TALLERES ESTUDIANTILES CIENCIAS UNAM
REAL ANALYSISSECOND EDITION
Edicin digital:Educacin
para todos
Edicin impresa:
Macmillan Second edition Stanford U.
Educacin
para todosEducacin para todos no es un proyecto lucrativo, sino un esfuerzo colectivo de estudiantes y profesores de la UNAM para facilitar el acceso a los materiales necesarios para la educacin de la mayor cantidad de gente posible. Pensamos editar en formato digital libros que por su alto costo, o bien porque ya no se consiguen en bibliotecas y libreras, no son accesibles para todos. Invitamos a todos los interesados en participar en este proyecto a sugerir ttulos, a prestarnos los textos para su digitalizacin y a ayudarnos en toda la labor tcnica que implica su reproduccin. El nuestro, es un proyecto colectivo abierto a la participacin de cualquier persona y todas las colaboraciones son bienvenidas. Nos encuentras en los Talleres Estudiantiles de la Facultad de Ciencias y puedes ponerte en contacto con nosotros a la siguiente direccin de correo electrnico: [email protected] http://eduktodos.org.mx
Contents
Prologue to the Student
1 Set Theory1 Introduction, 5 2 Functions, 8 3 Unions, intersections, and complements, 11 4 Algebras of sets, 16 5 The axiom of choice and infinite direct products, 18 6 Countable sets, 19 7 Relations and equivalences, 22 8 Partial orderings and the maximal principle, 23 9 Well ordering and the countable ordinals, 24
Part One
Theory of Functions of a Real Variable29
2 The Real Number System1 2 3 4 5 6 7
Axioms for the real numbers, 29 The natural and rational numbers as subsets of R, 32 The extended real numbers, 34 Sequences of real numbers, 35 Open and closed sets o j real numbers, 38 Continuousjunctions, 44 Bore1 sets, 50 ix
Contents
x i142
8 Topological Spaces1 Fundamental notions, 142 2 Bases and countability, 145 3 The separation axioms and continuous real-valuedfunctions, 147 4 Product spaces, 150 5 Connectedness, 150 "6 Absolute SS'S,154 "7 Nets, 155
9 Compact Spaces1 2 3 4 5 "6 7 "8
157
Basic properties, 157 Countable compactness and the Bolzano- Weierstrass property, 159 Compact metric spaces, 163 Products of compact spaces, 165 Locally compact spaces, 168 The Stone-eech compactijication, 170 The Stone- Weierstrass theorem, 171 The Ascoli theorem, 177
10 Banach Spaces1 Introduction, 181 2 Linear operators, 184 3 Linear.functionals and the Hahn-Banach theorem, 186 4 The closed graph theorem, 193 "5 Topological vector spaces, 197 "6 Weak topologies, 200 "7 Convexity, 203 8 Hilbert space, 210
181
Part Three
0
General Measure and Integration Theory217
1 Measure and Integration 11 2 3 "4 5 6 7
Measure spaces, 217 MeasurabIe.functions, 223 Integration, 225 General convergence theorems, 231 Signed measures, 232 The Radon-Nikodym theorem, 238 The L p spaces, 243
xii
Contents
12 Measure and Outer Measure1 Outer measure and measurability, 250 2 The extension theorem, 253 * 3 The Lebesgue-Stieltjes integral, 26 1 4 Product measures, 264 " 5 Inner measure, 274 "6 Extension by sets qf measure zero, 281 "7 Carathbodory outer measure, 283
250
13 The Daniel1 Integral1 2 3 4
286
Introduction, 286 The extension theorem, 288 Uniqueness, 294 Measurability and measure, 295
14 Measure and Topology1 2 3 "4
30 1
Baire sets and Borel sets, 301 Positive linear functionals and Baire measures, 304 Bounded 1inear.functionalson C ( X ) , 308 The Borel extension qf a measure, 313
15 Mappings of Measure Spaces1 Point mappings and set mappings, 317 2 Measure algebras, 319 3 Borel equivalences, 324 4 Set mappings andpoint mappings on complete metric spaces, 328 5 The isometries oj' L p , 331
317
Epilogue Bibliography Index of Symbols Subject Index
335 337 339 34 1
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