Lebesgue measure and integration

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Preface\0......Page 4
Contents\0......Page 8
1 Set And Set Inclusion\0......Page 10
2 Functions\0......Page 12
3 Supremum And Infimum\0......Page 14
4 Intervals\0......Page 15
5 Open, Closed .and Perfect Sets\0......Page 16
6 Sequences And Series\0......Page 18
7 Continuity And Differentiability\0......Page 21
1 Equivalent Sets\0......Page 24
3 Countable Sets\0......Page 28
4 Uncountable Sets\0......Page 37
5 Cardin.ality Of Sets\0......Page 42
6 Order Relation In Cardinal Numbers\0......Page 43
7 Addmon Of Cardinal Numbers\0......Page 46
8 Multiplication Of Cardinal Numbers\0......Page 47
9 Exponentiation Of Cardinal Numbers\0......Page 49
10 Cantor-like Sets\0......Page 53
1 Length Of Sets\0......Page 62
2 Outer Measure\0......Page 64
3 Lebesgue Measure\0......Page 72
4 Properties Of Measurable Sets\0......Page 74
5 Borel Sets And Their Measurability\0......Page 81
6 Further Properties Of Measurable Sets\0......Page 84
7 Characterizations Of Measurable Sets\0......Page 89
8 Nonmeasurable Sets\0......Page 92
1 Definition\0......Page 97
2 Properties Of Measurable Funcdons\0......Page 100
3 Step Function\0......Page 103
4 Operations On Measurable Funci\'ions\0......Page 104
5 Characteristic Funcfion\0......Page 109
6 Simple Function\0......Page 110
7 Continuous Funcfion\0......Page 112
8 Sets Of Measure Zero\0......Page 114
9. Borel Measurable Funcfion\0......Page 117
10 Sequence Of Functions\0......Page 119
11 The Structure Of Measurable Funcllons\0......Page 123
12. Convergence In Measure\0......Page 126
V Lebesgue Integral\0......Page 136
1 Riemann Integral\0......Page 137
2 Lebesgue Integral Of A Bounded Funci\'ion\0......Page 139
3 Comparison Of Riemann Integral And Lebesgue Integral\0......Page 145
4 Properties Of The Lebesgue Integral For Bounded Measurable Funct.ions\0......Page 147
5 Integral Of Nonnegative Measurable Func110ns\0......Page 153
6 General Lebesgue Integral\0......Page 163
7 Improper Integrals\0......Page 174
1 Dini Derivatives\0......Page 178
2 Differentiation Of Monotone Funci\'jons\0......Page 182
3 Functions Of Bounded Variation\0......Page 188
4 Differentiation Of An Integral\0......Page 194
5 Lebesgue Sets\0......Page 199
6 Absolutely Continuous Functions\0......Page 201
7 Integral Of The Derivative\0......Page 205
1 Notion Of Banach Spaces\0......Page 209
2 The Classes L^p\0......Page 214
3 The Holder And Minkowski Inequalities\0......Page 217
4 L^p Banach Spaces\0......Page 222
5 Convergence In The Mean\0......Page 225
6 Properties Of L^p Spaces\0......Page 227
7 Bounded Linear Ji\'unci\'ionals On L^p Spaces\0......Page 229
Appendix I: Existence Of Riemann Integral\0......Page 237
Appendix Ii: Nowhere Differentiable Continuous Functions\0......Page 241
Appendix Iii: The Development Of The Notion Of The Integral*\0......Page 244
Bibliography\0......Page 262
List Of Symbols And ~notations\0......Page 263
Index\0......Page 265