โ Scribed by J. C. Burkill
No coin nor oath required. For personal study only.
Dr Burkill gives a straightforward introduction to Lebesgue's theory of integration. His approach is the classical one, making use of the concept of measure, and deriving the principal results required for applications of the theory.
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
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Dr Burkill gives a straightforward introduction to Lebesgue's theory of integration. His approach is the classical one, making use of the concept of measure, and deriving the principal results required for applications of the theory.
Dr Burkill gives a straightforward introduction to Lebesgue's theory of integration. His approach is the classical one, making use of the concept of measure, and deriving the principal results required for applications of the theory.
