"This introduction to the theory of the Lebesgue Integral is primarily intended for third-year Honours students. A consistently geometrical approach has been chosen in order to show the simple underlying ideas. First, the volume of an n-dimensional set is defined as it s Lebesgue Measure. The integral of a non-negative function of n real variables is then the volume of its ordinate set; the extension to general functions follows easily. The old notion of the Riemann Integral is similarly developed; its relation to the Lebesgue Integral and the striking advantages of the latter are clearly set out."
The main object of this book is to provide an introduction to the theory of the so-called absolute integral. It is not an introduction to the "calculus": it is assumed that the reader is familiar with this. But the book should give the student a deeper understanding of the ideas underlying the calculus. It is also hoped that they will appreciate the aesthetic side of a purely mathematical theory, quite apart from its practical implications. Such an appreciation is quite as essential as technical skill.
Author(s): Werner W. Rogosinski
Series: University Mathematical Texts
Edition: 1
Publisher: Oliver and Boyd Edinburgh
Year: 1952
Language: English
Pages: 165
City: New York
Tags: Mathematical Analysis, Lebesgue integral, Riemann integral, Measure theory,
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
