Fundamental program of the calculus

Comments of the Reviewer......Page 9
History......Page 11
1.2. Bounds......Page 22
1.3. Sequences of real numbers......Page 23
1.6. Limits of monotonic sequences......Page 24
1.8. Limit superior and inferior......Page 25
1.9. Metric properties of sequence limits......Page 26
1.10. Series......Page 27
2.2. Open intervals......Page 30
2.4. Open sets......Page 31
2.5. Properties of open sets......Page 32
2.7. Closed sets......Page 34
2.8. Compact sets......Page 35
2.9. Compact interval......Page 36
3.2. Full covers and Cousin covers......Page 38
3.4. Cousin covering lemma......Page 40
3.5. Compactness arguments......Page 41
3.6. Covering arguments and measure theory......Page 43
3.7. Null sets......Page 45
3.8. Portions......Page 46
3.9. Language of meager/residual subsets......Page 47
4.1. Functions, increments, oscillations......Page 50
4.2. Growth of a function......Page 51
4.3. Full characterization of Lebesgue's measure......Page 52
4.4. Properties of the variation......Page 53
4.7. Continuity and absolute continuity......Page 54
4.8. Functions uniformly continuous throughout an interval......Page 55
4.9. Vitali's condition......Page 56
4.10. Continuous functions map compact intervals to compact intervals......Page 57
4.11. Discontinuities......Page 58
4.13. Convergent sequences map to convergent sequences......Page 59
4.14. Limits of continuous functions......Page 60
4.15. Local Continuity of limits......Page 61
4.16. Bernstein polynomials......Page 62
5.1. Growth of a function......Page 66
5.3. The Derivative......Page 67
5.4. Growth on a set......Page 68
5.6. Growth using upper or lower derivates......Page 69
5.7. Mean values......Page 70
6.1. Descriptive characterization of the integral......Page 72
6.3. Upper and lower integrals......Page 73
6.4. Integration of derivatives......Page 75
6.5. Estimates from Cauchy sums......Page 76
6.6. Estimates of integrals from derivates......Page 77
6.7. Elementary properties of the integral......Page 78
6.8. Summing inside the integral......Page 79
6.9. Integration of Null functions......Page 82
7.1. Null sets......Page 84
7.4. Almost closed sets......Page 85
7.6. Measure properties of derivates......Page 86
7.7. Measure computations with almost closed sets......Page 87
7.9. Measure estimates for almost closed sets......Page 89
7.10. Almost continuous functions......Page 90
7.11. Egorov-Taylor Theorem......Page 91
7.12. Characterizations of almost continuity......Page 95
8.1. Measure of compact sets......Page 98
8.2. Measure of almost closed sets......Page 99
8.4. Integral of nonnegative almost continuous functions......Page 100
8.6. Derivatives of monotonic functions......Page 101
8.7. The Lebesgue ``integral''......Page 102
8.8. Some measure estimates for Lebesgue's ``integral''......Page 103
8.9. Absolute continuity property of Lebesgue's ``integral''......Page 104
9.1. Integrability criteria......Page 106
9.2. Absolutely integrable functions......Page 108
9.4. Bounded, almost-continuous functions are absolutely integrable......Page 109
9.5. Henstock's integrability criterion......Page 110
9.7. Riemann's integrability criterion......Page 112
10.2. Theorem of Grace Chisolm Young......Page 114
10.3. Theorem of William Henry Young......Page 115
10.4. Theorem of Anthony P. Morse......Page 117
10.5. Measure properties of Dini derivatives......Page 118
10.6. Quasi-Cousin covers......Page 119
10.7. Quasi-Cousin Covering Lemma......Page 120
10.9. Estimates of integrals from Dini derivatives......Page 121
10.10. Growth lemmas on compact sets......Page 122
10.11. The Lebesgue differentiation theorem......Page 124
10.12. Differentiation of the integral......Page 125
11.1. Full and fine covers......Page 128
11.3. Variational Measures......Page 130
11.5. Measure properties of the variation......Page 131
11.8. Radó Covering Lemma......Page 132
11.9. Vitali Covering Theorem......Page 134
11.10. Fundamental Limit Theorems......Page 136
11.11. Fundamental theorem of the calculus......Page 137
11.12. Variational characterization of Lebesgue's measure......Page 138
11.13. The Density theorem......Page 139
11.14. Approximate continuity......Page 140
11.15. s-dimensional measures......Page 141
12.1. Integration of interval functions......Page 142
12.2. Henstock criterion......Page 143
12.4. Integrability of subadditive, continuous functions......Page 144
12.5. Jordan variation......Page 145
12.6. Jordan decomposition......Page 146
12.7. Absolute continuity in sense of Vitali......Page 147
12.8. Mutually singular functions......Page 149
12.9. Properties of the Jordan decomposition......Page 150
12.10. Singular functions......Page 151
12.11. Length of curves......Page 152
12.12. The Indicatrix......Page 154
13.1. Variational measures......Page 156
13.2. Variational estimates......Page 157
13.3. Lipschitz numbers......Page 158
13.5. Variational classifications of real functions......Page 159
13.6. Local behaviour of functions......Page 161
13.7. Derivates and variation......Page 162
13.9. Functions having -finite full variation......Page 164
13.10. Variation on compact sets......Page 166
13.12. Vitali property and differentiability......Page 167
13.13. Differentiability properties from the Vitali property......Page 168
13.14. The Vitali property and variation......Page 169
13.15. Characterization of the Vitali property......Page 170
13.17. Mapping properties......Page 171
13.18. Lusin's conditions......Page 172
13.19. Banach-Zarecki Theorem......Page 173
14.1. Rudimentary properties of the integral......Page 176
14.2. Properties of the indefinite integral......Page 179
14.3. McShane's criterion and the absolute integral......Page 180
14.5. Expression of the integral as a measure......Page 182
14.6. Riemann's criterion......Page 183
14.7. Freiling's criterion......Page 185
14.8. Limits of integrable functions......Page 186
14.9. Local absolute integrability conditions......Page 189
14.11. A characterization of the integral......Page 192
15.1. Reduction theorem......Page 196
15.2. Variational properties......Page 197
15.3. Derivative of the integral......Page 198
15.5. Helley's first theorem......Page 199
15.7. Linear functionals......Page 201
15.8. Representation of positive linear functionals on C[a,b]......Page 203
15.10. Hellinger integrals......Page 205
15.11. Bounded linear functionals on AC0[a,b]......Page 207
APPENDIX: Formal Theory of the Calculus......Page 208
15.13. Functions defined on covering relations......Page 210
15.16. Differentiation bases......Page 211
15.18. Properties of full/fine covers......Page 212
15.20. The measures......Page 213
15.22. Kolmogorov equivalence......Page 214
15.24. Integrability of subadditive, continuous functions......Page 215
15.27. Rudimentary properties of the integral......Page 216
15.28. Henstock's criterion......Page 217
15.30. Limits......Page 218
15.32. Kolmogorov equivalence from a limit......Page 220
15.35. The fundamental theorem of the calculus......Page 221
Afterword......Page 222
Index......Page 224