Book:J.C. Burkill/The Lebesgue Integral

Subject Matter

 * The Lebesgue Integral

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



Source work progress
* : Chapter $\text {I}$: Sets of Points: $1 \cdot 1$. The algebra of sets