An Introduction to the Theory of Real Functions

Title page......Page 1
CONTENTS......Page 3
Preface......Page 7
Introduction......Page 9
0.1 Infima and Suprema......Page 10
0.2 Limits......Page 12
0.3 The Diagonal Principle of Choice......Page 16
Notes......Page 17
1.1 Monotone Functions......Page 18
1.2 Sequences of Monotone Functions......Page 20
1.3 The Variation of a Function......Page 22
The Relationship with Arc-length......Page 25
1.4 Jordan Canonical Decomposition......Page 26
1.5 The Riemann-Stieltjes Integral. Existence Criteria......Page 28
1.6 Properties of the Integral......Page 34
Taking the Limit under the Integral Sign......Page 37
Notes......Page 38
2.2 Tonelli Polynomials......Page 39
2.3 Bernstein Polynomials......Page 41
Application to Absolutely Monotone Functions......Page 43
2.4 The Stone-Weierstrass Theorem......Page 47
Notes......Page 51
3.1 Continuous Functions......Page 53
Uniform and Continuous Convergence......Page 55
3.2 Equicontinuous Families of Functions......Page 57
3.3 Semicontinuous Functions......Page 59
3.4 Maximum and Minimum at a Point......Page 63
3.5 Functions of the First Class of Baire......Page 65
4.1 Algebras of Sets......Page 73
4.2 Cartesian Products......Page 76
4.3 Borel Sets......Page 77
Classification of Borel Sets......Page 79
4.4 Measurable Functions......Page 81
4.5 Baire Functions......Page 86
Notes......Page 88
5.1 Measure......Page 89
Sets of Measure Zero......Page 90
5.2 Outer Measure......Page 93
Extension of a Finitely Additive Function to a Measure......Page 99
5.3 Content of an Interval......Page 100
5.4 Lebesgue Measure......Page 104
A Characterization of the Modulus of a Determinant......Page 112
Non-Measurable Sets......Page 115
5.5 Measurable Functions......Page 116
L_n-measurable Functions......Page 117
5.6 Sequences of Measurable Functions......Page 119
Notes......Page 123
6.1 Integration of Non-negative Functions......Page 125
The Banach-Vitali Theorem......Page 131
6.2 Integration of Functions of Arbitrary Sign. Summability......Page 132
Relationship of Lebesgue Measure and Integral with the Riemann Integral......Page 138
6.3 Cartesian Product of Measures. Fubini's Theorem......Page 140
The case of Lebesgue Measure and Integrals......Page 144
6.4 Riesz's Theorem......Page 147
Proof of Riesz's Theorem......Page 148
Uniqueness of Measure in Riesz's Theorem......Page 150
Notes......Page 151
7.1 Differentiability Almost Everywhere......Page 153
Rademacher's Theorem......Page 160
7.2 Interval Functions of Bounded Variation......Page 164
7.3 Absolutely Continuous Interval Functions......Page 168
7.4 The case of Functions of a Single Variable......Page 171
Arc Length......Page 178
Dini Derivatives......Page 180
Non-differentiable Continuous Functions......Page 184
7.5 Countably Additive Set Functions......Page 186
Riesz's Theorem......Page 191
Connection with Interval Functions......Page 193
Lebesgue-Stieltjes Integral......Page 198
7.6 The Radon-Nikodym Theorem......Page 200
Change of Variable in a Lebesgue Integral......Page 207
Notes......Page 209
Appendix 1 Another Proof of Lebesgue's Theorem on the Differentiability of the Integral......Page 212
Appendix 2 Another Proof of Fubini's Theorem for Lebesgue Integrals......Page 213
Appendix 3 Stepanov's Theorem on Differentiability Almost Everywhere......Page 216
Appendix 4 Proofs of the Sierpinski-Young and Denjoy-Young-Saks Theorems......Page 218
Exercises (Compiled by M. Kosiek, W. Mlak and Z. Opial.)......Page 219
Bibliography......Page 234
Subject Index......Page 235