This textbook leads the reader by easy stageshroug t h the essential
parts of the theory of sets and theory of measure to the properties
of the Lebesgue integral. The first part of the book gives a general
introduction to functions of a real variable, measure, and
integration, while the second part treats the problem of inverting
the derivative of continuous functions, leading to the Denjoy
integrals, and studies the derivates and approximate derivates of
functions of a real variable on arbitrary linear sets. The author
considers the presentation of this second part as the main purpose
of his book.
H. L. JEFFERY, who holds degrees from Acadia University and
Cornell University, was for many years Professor of Mathematics
and Head of the Department of Mathematics at Queens University,
Kingston. He is now Professor of Mathematics at Acadia University,
Wolfville, Nova Scotia. He is the author of other important works
in mathematics, Calculus and Trigonometric Series: A Survey,
published by the University of Toronto Press
Author(s): Ralph Lent Jeffery
Series: Mathematical Expositions; 6
Edition: 2
Publisher: University of Toronto Press
Year: 1951
Language: English
Pages: 246
Tags: Theory of Functions, Lebesgue Integral
0. INTRODUCTION
0.1. The positive integers 3
0.2. The fundamental operations on integers 4
0.3. The rational numbers 7
0.4. The irrational numbers 9
0.5. The real number system 12
Problems 18
1. SETS, SEQUENCES, AND FUNCTIONS
1.1. Bounds and limits of sets and sequences 20
1.2. Functions and their properties 30
1.3. Sequences of functions and uniform convergence 36
Problems 41
II. METRIC PROPERTIES OF SETS
2.1. Notation and definitions 45
2.2. Descriptive properties of sets 46
2.3. Metric properties of sets 48
2.4. Measurability and measurable sets 54
2.5. Further descriptive properties of sets 60
2.6. Measure-preserving transformations and non-measurable sets 61
2.7. A non-measurable set 62
Problems 64
III. THE LEBESGUE INTEGRAL
3.1. Measurable functions 66
3.2. The Lebesgue integral 67
3.3. The Riemann integral 69
3.4. The extension of the definition of the Lebesgue
integral to unbounded functions 73
3.5. Further properties of measurable functions 76
Problems 78
IV. PROPERTIES OF THE LEBESGUE INTEGRAL
4.1. Notation and conventions 81
4.2. Properties of the Lebesgue integral 81
4.3. Definitions of summability and their extension to unbounded sets 88
4.4. The integrability of sequences 92
4.5. Integrals containing a parameter 95
4.6. Further theorems on sequences of functions 97
4.7. The ergodic theorem 100
Problems 106
V. METRIC DENSITY AND FUNCTIONS OF BOUNDED VARIATION
5.1. The Vitali covering theorem 110
5.2. Metric density of sets 114
5.3. Approximate continuity 118
5.4. Functions of bounded variation 118
5.5. Upper and lower derivatives 122
5.6. Functions of sets 125
5.7. The summability of the derivative of a function of bounded variation 127
5.8. Functions of sets 131
Problems 137
VI. THE INVERSION OF DERIVATIVES
6.1. Functions defined by integrals, F (x) = L(f,a,x) 140
6.2. The inversion of derivatives which are not summable 146
6.3. The integrals of Denjoy and other generalized integrals 158
6.4. Descriptive definitions of generalized integrals 159
Problems 162
VII. DERIVED NUMBERS AND DERIVATIVES
7.1. Derivatives or derived numbers 165
7.2. The Weierstrass non-differentiable function 168
7.3. A function which has no unilateral derivative 172
7.4. The derived numbers of arbitrary functions defined on arbitrary sets 181
7.5. Approximate derived numbers over arbitrary sets 187
7.6. Approximate derived numbers of measurable functions, and relations between arbitrary functions
and measurable functions 199
VIII. THE STIELTJES INTEGRAL
8.1. The Riemann—Stieltjes Integral 204
8.2. Properties of the Riemann-Stieltjes integral 205
8.3. Interval functions and measure functions 211
8.4. Linear functionals 212
BIBLIOGRAPHY 221
INDEX OF SUBJECTS 225
INDEX OF AUTHORS 231
