Volume and integral

Title page
PREFACE
PART I: VOLUME
CHAPTER 1 SETS OF POINTS
1. The Euclidean space
2. The space of n dimensions
3. Sets
4. Subsets
5. Complements
6. Products
7. Sums
8. Sequences
9. Enumerable sets
10. The rational set
11. Non-enumerable sets
12. Interior, exterior, and frontier of a set
13. Open and closed sets
14. Limiting points
15. Perfect sets
16. Borel's covering theorem
17. Functions of real variables
CHAPTER II CONTENT
1. The problem of volume
2. Postulates
3. Interval sums
4. Outer content of bounded sets
5. Outer content (general)
6. Inner content
7. Content
8. Content of elementary sets
9. Deficiencies
CHAPTER III MEASURE
1. A new postulate
2.. Interval sets
3. Volume of interval sets
4. Outer measure
5. Inner measure of bounded sets
6. Inner measure (general)
7. Measure
8. Limiting properties of Ineasure
9. Non-measurable sets
10. Vitali's covering theorem
PART II: THE INTEGRAL
CHAPTER IV RIEMANN'S INTEGRAL
1. Volume and integral
2. Lower and upper integrals
3. Riemann's integral
4. Riemann's sums
5. Deficiencies of the R-integral
CHAPTER V LEBESGUE'S INTEGRAL
1. Lower and upper L-integrals
2. Lebesgue's integral
3. Limiting theorems
4. Measurable functions
5. Integrable functions
6. Lebesgue's sums
7. Fubini's theorem
8. Relation between R-integral and L-integral
CHAPTER VI INTEGRATION AND DIFFERENTIATION
1. The problem
2. Elementary properties of the indefinite integral
3. The second mean- value theorem
4. Functions of bounded variation
5. The derivative of the indefinite integral
6. Integration of a derivative
7. Integration by parts and rule of substitution
FURTHER LITERATURE ON THE INTEGRAL
INDEX