The Lebesgue Integral

Title......Page 2
Copyright Page......Page 3
Preface......Page 4
Contents......Page 6
The algebra of sets......Page 8
Infinite sets......Page 10
Sets of points. Descriptive properties......Page 11
Covering theorems......Page 13
Plane sets......Page 14
Measure of open sets......Page 17
Measure of closed sets......Page 18
Open and closed sets......Page 19
Outer and inner measure. Measurable sets......Page 20
The additive property of measure......Page 21
Non-measurable sets......Page 22
Further properties of measure......Page 23
Sequences of sets......Page 25
Plane measure......Page 28
Measurable functions......Page 30
The Lebesgue integral......Page 33
The Riemann integral......Page 34
The scope of Lebesgue's definition......Page 35
The integral as the limit of approximative sums......Page 37
The integral of an unbounded function......Page 38
The integral over an infinite range......Page 40
Simple properties of the integral......Page 41
Sets of measure zero......Page 44
Sequences of integrals of positive functions......Page 45
Sequences of integrals (integration term by term)......Page 47
Change of variable......Page 65
The derivates of a function......Page 51
Vitali's covering theorem......Page 53
Differentiability of a monotonic function......Page 55
The integral of the derivative of an increasing function......Page 56
Functions of bounded variation 50 Differentiation of the indefinite integral......Page 59
Absolutely continuous functions......Page 61
The metric space L^p......Page 68
Fubini's theorem......Page 70
The class L^p......Page 72
Integration with respect to a function......Page 77
The variation of an increasing function......Page 78
The Lebesgue·Stieltjes integral......Page 79
Integration by parts......Page 82
Change of variable. Second mean·value theorem......Page 84
Solutions of some examples......Page 87