FUNDAMENTALS OF REAL ANALYSIS

Transcription

1 ÜO i FUNDAMENTALS OF REAL ANALYSIS James Foran University of Missouri Kansas City, Missouri Marcel Dekker, Inc. New York * Basel - Hong Kong

2 Preface iii 1.1 Basic Definitions and Background Material 1 Set notation 1 Natural, rational, and real numbers 5 Euclidean Spaces and vector spaces 12 Inner products and norms 13 Metric Spaces and topological spaces 15 Topological properties: Hausdörff, regulär, normal, compact, locally compact 18 Compact metric spaces, Heine-Borel theorem 23 Complete metric spaces 25 The Baire category theorem 29 Functions, limits and continuity 31 Uniform limits and uniform continuity 33 Riemann and Riemann-Stieltjes integrals 34 Exercises 35 VII

3 viii 1.2 Additional Basic Theorems 38 Continuous image of compact sets 38 The sup norm and C(X) 39 Nowhere differentiable functions are residual 41 Examples of nowhere differentiable functions 43 Connected sets and Darboux functions 47 Uniform limits of Riemann-Stieltjes integrals 52 Uniform limits of derivatives 53 Weierstrass approximation theorem 55 Exercises Sets and Cardinal Numbers 58 Cardinal equivalence 59 The Cantor-Bernstein theorem 61 Countable sets 64 The set of subsets of a set 67 Discussion of paradoxes 68 Cardinal arithmetic 70 Characterization of open subsets of the reals 75 Characterization of closed subsets of the reals 76 Some sets of cardinality c 77 Sets of points of discontinuity of a function 78 Exercises Discussion of Axiomatic Set Theory 85 Logic and the formulas of set theory 85 The ZF axiom 87 Brief discussion of modeis 90 The elements of Real Analysis in ZFC 91 Exercises Well-Ordered Sets and Ordinal Numbers 94 Order Types 17, w, X Similarity of bounded perfect sets Well-ordered sets and ordinal numbers The well-ordering theorem Comparability of cardinal numbers 114 Definition by transfinite induction 116 Proof by transfinite induction; the sum and product of cardinals 117 The continuum problem and Souslin's problem 119 The derived subsets of a closed set 122 Examples constructed by transfinite induction 123 Exercises 126

4 .' ix 3.2 Applications of the Axiom of Choice 127 Zorn's lemma and the maximality principle 128 Bases for linear spaces 130 Discontinuous linear functionals 132 Hahn-Banach theorem 135 Hausdörff paradoxical theorem 138 Exercises Borel Sets and Baire Functions 142 Definition of Borel sets and Baire functions 143 Borel classes defined üsing ordinal numbers 144 Lower level Borel classes 144 Inclusion relationships between the classes 146 Cardinality of the Borel subsets of the line 149 Connection between Borel sets and logic 151 Baire classes defined using ordinal numbers 153 Derivatives and semi-continuous functions 155 The characteristic functions of Borel sets 158 Borel sets and Baire functions 159 Exercises Exact Baire Classes 166 Points of continuity of Baire 1 functions 167 Baire classes closed under uniform limits 169 Relation between Baire classes and Borel sets 171 Characterizations of Baire 1 functions 174 Composition of Baire functions 175 Existence of universal Baire functions 176 Borel classes are strictly increasing 178 Exercises Measure and Measurable Sets 180 Description of measure and outer measure 181 Caratheodory's theorem 186 Complete measures 190 Examples of measures 193 Lebesgue measure and measurability of sets 194 Borel regularity 197 Product measures 200 The Vitali covering theorem 204 The Lebesgue density theorem 2 08 Exercises 210

5 5.2 Additive Functions of a Set 212 Completely additive functions of a set 212 The Jordan decomposition 216 The Hahn decomposition 218 Hausdörff s-dimensional measure 221 Metrie outer measures 222 Borel regularity 225 Measures generated by nets 228 Additive functions of an interval 230 Probability measures, examples 241 Exercises Measurable Functions 250 Definition and equivalent definitions 251 Sums, products, limits of measurable functions 255 Egoroff's theorem 256 Lusin's theorem 258 Equivalence to Baire 2 functions a.e Conditions for Riemann integrability 2 64 A derivative which is not Riemann integrable 267 Exercises Density Topology and Approximate Continuity 270 Approximate continuity and approximate limits 270 An equivalent definition 272 The density topology 276 Lusin-Menchoff theorem 281 The density topology is not normal 283 Notes on further density results 285 Exercises The Lebesgue Integral 288 Discussion of the integral 288 The integral of bounded functions 294 Lebesgue integral compared to Riemann integral 298 The integral of non-negative functions 299

6 i XI The general Lebesgue integral 303 Alternate approaches 305 Properties of product measures 308 Fubini's theorem 310 Measurability of sets in product measures 314 Improper integrals 319 Exercises Introduction to X? Spaces 321 Conjugate functions and the spaces 2? 322 Holder's inequality and Minkowski's ineguality 325 Convergence in mean and in measure 327 Completeness, Riesz-Fisher theorem 334 Random variable and their distributions 336 The integral applied to expected value 338 Exercises Differentiation of the Integral 343 Functions of an interval 344 Derivates of a function 346 Differentiability a.e. of monotone functions 348 Bounded Variation, absolute continuity 350 Characterization of the Lebesgue integral 353 The Lebesgue decomposition 357 Variation and arc length of the graph 358 Lipschitz functions 362 Convex functions 3 63 Exercises Differentiation of Functions of a Set 367 Absolute continuity 3 67 Radon-Nikodym theorem 370 The Radon-Nikodym derivative 373 Singular functions, Lebesgue decomposition 377 Derivates in Euclidean spaces 384 Differentiation of additive functions of a set 385 Linear functionals in general setting 395 The Riesz representation theorem for X? 399 Exercises 402

7 The Denjoy-Perron Integral 404 General notion of an integral 406 Cauchy and Harnack extensions of an integral 409 Constructive definition of Denjoy 411 Functions AC* and BV* on a set E 412 The V*-integral 420 Majorants and minorants 426 The Perron integral 427 Equivalence of the Perron and E*-integral 434 The Riemann-Complete integral 441 Equivalence to the Denjoy-Perron integral 441 Exercises 443 The Denjoy Integral 445 Notes on the Perron integral in K 445 Functions AC and BV on a set E 446 Properties of ACG and BVG functions 450 Monotonicity theorems 453 The constructive definition 457 Equivalence to the Z)-integral 458 Criteria for the classes ACG and ACG* 461 Notes on more general integrals 465 Exercises 467 Index 469