The Lebesgue Integral

Contents......Page 5
Preface......Page 3
1-1 The algebra of sets......Page 7
1-2 Infinite sets......Page 9
1-3 Sets of points. Descriptive properties......Page 10
1-4 Covering theorems......Page 12
1-6 Plane sets......Page 13
2-2 Measure of open sets......Page 16
2-3 Measure of closed sets......Page 17
2-4 Open and closed sets......Page 18
2-6 Outer and inner measure. Measurable sets......Page 19
2-6 The additive property of measure......Page 20
2-7 Non-measurable sets......Page 21
2-8 Further properties of measure......Page 22
2-9 Sequences of sets......Page 24
2-10 Plane measure......Page 27
2-12 Measurable functions......Page 29
3-1 The Lebesgue integral......Page 32
3-2 The Riemann integral......Page 33
3-3 The scope of Lebesgue's definition......Page 34
3-4 The integral as the limit of approximative sums......Page 36
3-5 The integral of an unbounded function......Page 37
3-6 The integral over an infinite range......Page 39
3-7 Simple properties of the integral......Page 40
3-8 Sets of measure zero......Page 43
3-9 Sequences of integrals of positive functions......Page 44
3-10 Sequences of Integrals (Integration Term by Term)......Page 46
4-2 The derivates of a function......Page 50
4-3 Vitali's covering theorem......Page 52
4-4 Differentiability of a monotonie function......Page 54
4-5 The integral of the derivative of an increasing function......Page 55
4-6 Functions of bounded variation......Page 56
4-7 Differentiation of the indefinite integral......Page 58
4-8 Absolutely continuous functions......Page 60
5-2 Change of variable......Page 64
5-3 Multiple integrals......Page 67
5-4 Fubini's theorem......Page 69
5-6 The class L^p......Page 71
5-7 The metric space L^p......Page 73
6-1 Integration with respect to a function......Page 76
6-2 The variation of an increasing function......Page 77
6-3 The Lebesgue-Stieltjes integral......Page 78
6-4 Integration by parts......Page 81
6-5 Change of variable. Second mean-value theorem......Page 83
Solutions of some examples......Page 86