Volume and Integral [15, 1 ed.]

"This introduction to the theory of the Lebesgue Integral is primarily intended for third-year Honours students. A

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English Pages 165 Year 1952

Author / Uploaded
Werner W. Rogosinski

Categories
Mathematics

Table of contents :
PREFACE v

PART I - VOLUME

CHAPTER I - SETS OF POINTS

1. The Euclidean space 3
2. The space of n dimensions 3
3. Sets 4
4. Subsets 6
5. Complements 7
6. Products 7
7. Sums 8
8. Sequences 9
9. Enumerable sets 12
10. The rational set 13
11. Non-enumerable sets 14
12. Interior, exterior, and frontier of a set 15
13. Open and closed sets 16
14. Limiting points 18
15. Perfect sets 21
16. Borel’s covering theorem 24
17. Functions of real variables 25

CHAPTER II - CONTENT

1. The problem of volume 31
2. Postulates 32
3. Interval sums 33
4. Outer content of bounded sets 35
5. Outer content (general) 38
6. Inner content 38
7. Content 40
8. Content of elementary sets 44
9. Deficiencies 47

CHAPTER III - MEASURE

1. A new postulate 5O
2. Interval sets 50
3. Volume of interval sets 52
4. Outer measure 55
5. Inner measure of bounded sets 58
6. Inner measure (general) 62
7. Measure 63
8. Limiting properties of measure 64
9. Non-measurable sets 68
10. Vitali’s covering theorem 69

PART II—THE INTEGRAL

CHAPTER IV - RIEMANN’S INTEGRAL

1. Volume and integral 75
2. Lower and upper integrals 79
3. Riemann’s integral 81
4. Riemann’s sums 86
5. Deficiencies of the R-integral 94

CHAPTER V - LEBESGUE’S INTEGRAL

1. Lower and upper L-integrals 97
2. Lebesgue’s integral 99
3. Limiting theorems 102
4. Measurable functions 106
5. Integrable functions 110
6. Lebesgue’s sums 117
7. Fubini’s theorem 121
8. Relation between R-integral and L-integral 127

CHAPTER VI - INTEGRATION AND DIFFERENTIATION

1. The problem 132
2. Elementary properties of the indefinite integral 133
3. The second mean-value theorem 134
4. Functions of bounded variation 136
5. The derivative of the indefinite integral 143
6. Integration of a derivative 146
7. Integration by parts and rule of substitution 153

FURTHER LITERATURE ON THE INTEGRAL 157

INDEX 159