This paperback, which comprises the first part of Introduction to Measure and Probability by J. F. C. Kingman and S. J. Taylor, gives a self-contained treatment of the theory of finite measures in general spaces at the undergraduate level. It sets the material out in a form which not only provides an introduction for intending specialists in measure theory but also meets the needs of students of probability. The theory of measure and integration is presented for general spaces, with Lebesgue measure and the Lebesgue integral considered as important examples whose special properties are obtained. The introduction to functional analysis which follows covers the material to probability theory and also the basic theory of L2-spaces, important in modern physics. A large number of examples is included; these form an essential part of the development.
Author(s): S. J. Taylor
Publisher: University Press
Year: 1973
Language: English
Pages: 273
City: Cambridge [Eng.]
Cover......Page 1
Title......Page 2
Copyright......Page 3
CONTENTS ......Page 4
Preface ......Page 6
1.1 Sets ......Page 8
1.2 Mappings ......Page 10
1.3 Cardinal numbers ......Page 12
1.4 Operations on subsets ......Page 16
1.5 Classes of subsets ......Page 21
1.6 Axiom of choice ......Page 26
2.1 Metric space ......Page 30
2.2 Completeness and compactness ......Page 36
2.3 Functions ......Page 42
6.2 Product measures 1. ......Page 45
2.5 Further types of subset ......Page 48
2.6 Normed linear space ......Page 51
2.7 Cantor set ......Page 56
3.1 Types of set function ......Page 58
3.2 Hahn-Jordan decompositions ......Page 68
3.3 Additive set functions on a ring ......Page 72
3.4 Length, area and volume of elementary figures ......Page 76
4.1 Extension theorem ; Lebesgue measure ......Page 81
4.2 Complete measures ......Page 88
4.3 Approximation theorems ......Page 91
4.4* Geometrical properties of Lebesgue measure ......Page 95
4.5 Lebesgue-Stieltjes measure ......Page 102
5.1 What is an integral? ......Page 107
5.2 Simple functions; measurable functions ......Page 108
5.3 Definition of the integral ......Page 117
5.4 Properties of the integral ......Page 122
5.5 Lebesgue integral; Lebesgue-Stieltjes integral ......Page 131
5.6* Conditions for integrability ......Page 134
6.1 Classes of subsets in a product space page ......Page 141
6.3 Fubini's theorem ......Page 150
6.4 Radon-Nikodym theorem ......Page 155
6.5 Mappings of measure spaces ......Page 160
6.6* Measure in function space ......Page 164
6.7 Applications ......Page 169
7.1 Point-wise convergence ......Page 173
7.2 Convergence in measure ......Page 178
7.3 Convergence in pth mean ......Page 181
7.4 Inequalities ......Page 190
7.5* Measure preserving transformations from a space to itself ......Page 194
8.1 Dependence of 22 on the underlying (S, , ,a) ......Page 201
8.2 Orthogonal systems of functions ......Page 206
8.3 Riesz-Fischer theorem ......Page 209
8.4* Space of linear functionals ......Page 216
8.5* The space conjugate to Y. ......Page 222
8.6* Mean ergodic theorem ......Page 226
9.1 Differentiating a monotone function ......Page 231
9.2 Differentiating the indefinite integral ......Page 237
9.3 Point-wise differentiation of measures ......Page 243
9.4* The Daniell integral ......Page 248
9.5* Representation of linear functionals ......Page 257
9.6* Haar measure ......Page 261
Index of notation ......Page 268
General Index ......Page 270
