General theory of functions and integration

1. The Real Numbers. Point Sets and Sequences
1-0 Introduction 1
1-1 The real line 2
1-2 Notation and terminology concerning sets 7
1-3 Construction of a complete ordered field 12
1-4 Archimedean order. Countability of the rationals 16
1-5 Point sets. Upper and lower bounds 21
1-6 Real sequences 24
1-7 The extended real number system 29
1-8 Cauchy’s convergence condition 32
1-9 The theorem on nested intervals 34

2. Euclidean Space. Topology and Continuous Functions
2-0 Introduction 37
2-1 The space Rᵏ 38
2-2 Linear configurations in Rᵏ 43
2-3 The topology of Rᵏ 47
2-2-4 Nests, points of accumulation, and convergent sequences 56
2-5 Covering theorems 61
2-6 Compactness 65
2-7 Functions. Continuity 68
2-8 Connected sets 76
2-9 Relative topologies 83
2-10 Cantor’s ternary set 86

3. Abstract Spaces
3-0 Introduction 89
3-1 Topological spaces 90
3-2 Compactness and other properties 96
3-3 Postulates of separation 102
3-4 Postulated neighborhood systems 105
3-5 Compactification. Local compactness 107
3-6 Metric spaces 111
3-7 Compactness in metric spaces 121
3-8 Completeness and completion 124
3-9 Category 127
3-10 Zorn’s lemma 131
3-11 Cartesian product topologies 134
3-12 Vector spaces 140
3-13 Normed linear spaces 144
3-14 Hilbert spaces 153
3-15 Spaces of continuous functions 164

4. The Theory of Measure
4-0 Introduction 177
4-1 Algebraic operations in R* 178
4-2 Rings and σ-Rings 180
4-3 Additive set functions 185
4-4 Some properties of measures 188
4-5 Preliminary remarks about Lebesgue measure 189
4-6 Lebesgue outer measure in Rᵏ 191
4-7 The measure induced by an outer measure 196
4-8 Lebesgue measure in Rᵏ 203
4-9 A general method of constructing outer measures 215
4-10 Outer measures in R from monotone functions 218
4-11 The completion of a measure 224

5. The Lebesgue Integral
5-0 Introduction 226
5-1 Measurable functions 229
5-2 The integral of a bounded function 240
5-3 Preliminary convergence theorems 248
5-4 The general definition of an integral 252
5-5 Some basic convergence theorems 260
5-6 Convergence in measure 265
5-7 Convergence in mean 270
5-8 The Lᵖ spaces 274
5-9 Integration with respect to the completion of a measure 278

6. Integration by the Daniell Method
6-0 Introduction 281
6-1 Elementary integrals on a vector lattice of functions 282
6-2 Over-functions and under-functions 284
6-3 Summable functions 288
6-4 Sets of measure zero 292
6-5 Measurable functions and measurable sets 297
6-6 The N-norm 300
6-7 Connections with Chapter 5 303
6-8 Induction of Lebesgue measure in Rᵏ 310
6-9 Arbitrary elementary integrals on C°°(Rᵏ) 313
6-10 Regular Borel measures in Rᵏ 319
6-11 The class ℒᵖ 321

7. Iterated Integrals and Fubini’s Theorem
7—0 Introduction 324
7—1 The Fubini theorem for Euclidean spaces 326
7—2 The Fubini-Stone theorem 329
7~3 Products of functions of one variable 334
7—4 Iterated integrals and products of Euclidean spaces 336
7—5 Abstract theory of product-measures 339
7-6 The abstract Fubini theorem 345

8. The Theory of Signed Measures
8—0 Introduction 348
8—1 Signed measures 349
8—2 Absolute continuity 356
8—3 The Radon-Nikodym theorem 358
8—4 Continuous linear functionals on Lᵖ(μ) 361
8—5 The Lebesgue decomposition of a signed measure 364
8-6 Alternative approach via elementary integrals 366
8-7 The decomposition of linear functionals 371

9. Functions of One Real Variable
9-0 Introduction 379
9-1 Monotone functions 379
9-2 Vector-valued functions of bounded variation 382
9-3 Rectifiable curves 388
9-4 Real-valued functions of bounded variation 390
9-5 Stieltjes integrals 392
9-6 Convergence theorems for functions of bounded variation 398
9-7 Differentiation of monotone functions 402
9-8 Absolutely continuous functions 410

BIBLIOGRAPHY 423

LIST OF SPECIAL SYMBOLS 429

INDEX 431