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

## J.C. Burkill: *The Lebesgue Integral*

Published $1951$, **Cambridge Tracts in Mathematics and Mathematical Physics No. 40 (Cambridge University Press)**.

### Subject Matter

### 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