# Book:J.C. Burkill/The Lebesgue Integral

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## J.C. Burkill: The Lebesgue Integral

Published $\text {1951}$, Cambridge Tracts in Mathematics and Mathematical Physics No. 40 (Cambridge University Press).

### Contents

Author's Preface
Chapter I. SETS OF POINTS
1.1 The algebra of sets
1.2 Infinite sets
1.3 Sets of points. Descriptive properties
1.4 Covering theorems
1.5 Plane sets
Chapter II. MEASURE
2.1 Measure
2.2 Measure of open sets
2.3 Measure of closed sets
2.4 Open and closed sets
2.5 Outer and inner measure. Measurable sets
2.6 The additive property of measure
2.7 Non-measurable sets
2.8 Further properties of measure
2.9 Sequences of sets
2.10 Plane measure
2.11 Measurability in the sense of Borel
2.12 Measurable functions
Chapter III. THE LEBESGUE INTEGRAL
3.1 The Lebesgue integral
3.2 The Riemann integral
3.3 The scope of Lebesgue's definition
3.4 The integral as the limit of approximating sums
3.5 The integral of an unbounded function
3.6 The integral over an infinite range
3.7 Simple properties of the integral
3.8 Sets of Measure zero
3.9 Sequences of integrals of positive functions
3.10 Sequences of integrals (integration term by term)
Chapter IV. DIFFERENTIATION AND INTEGRATION
4.1 Differentiation and integration as inverse processes
4.2 The derivatives of a function
4.3 Vitali's covering theorem
4.4 Differentiability of a monotonic function
4.5 The integral of the derivative of an increasing function
4.6 Functions of bounded variation
4.7 Differentiation of the indefinite integral
4.8 Absolutely continuous functions
Chapter V. FURTHER PROPERTIES OF THE INTEGRAL
5.1 Integration by parts
5.2 Change of variable
5.3 Multiple integrals
5.4 Fubini's theorem
5.5 Differentiation of multiple integrals
5.6 The class $L^p$
5.7 The metric space $L^p$
Chapter VI. THE LEBESGUE-STIELTJES INTEGRAL
6.1 Integration with respect to a function
6.2 The variation of an increasing function
6.3 The Lebesgue-Stieltjes integral
6.4 Integration by parts
6.5 Change of variable. Second mean-value theorem
Solutions of some examples

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