Hausdorff measures

C. A. Rogers9780521079709, 0521079705

When originally published, this text was the first general account of Hausdorff measures, a subject that has important applications in many fields of mathematics. The first of the three chapters contains an introduction to measure theory, paying particular attention to the study of non-sigma-finite measures. The second chapter develops the most general aspects of the theory of Hausdorff measures, and the final chapter gives a general survey of applications of Hausdorff measures followed by detailed accounts of two special applications. This new edition has a foreword by Kenneth Falconer outlining the developments in measure theory since this book first appeared. This book is ideal for graduate mathematicians with no previous knowledge of the subject, but experts in the field will also want a copy for their shelves

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Table of contents :
Title page ……Page 1
Date-line ……Page 2
CONTENTS ……Page 3
Preface ……Page 5
2 Measures in abstract spaces ……Page 7
3 Measures in topological spaces ……Page 28
4 Measures in metric spaces ……Page 32
5 Lebesgue measure in $n$-dimensional Euclidean space ……Page 46
6 Metric measures in topological spaces ……Page 49
7 The Souslin operation ……Page 50
1 Definition of Hausdorff measures and equivalent definitions ……Page 56
2 Mappings, special Hausdorff measures, surface areas ……Page 59
3 Existence theorems ……Page 64
4 Comparison theorems ……Page 84
5 Souslin sets ……Page 90
6 The increasing sets lemma and its consequences ……Page 96
7 The existence of comparable net measures and their properties ……Page 107
8 Sets of non-$sigma$-finite measure ……Page 129
1 A survey of applications of Hausdorff measures ……Page 134
2 Sets of real numbers defined in terms of their expansions into continued fractions ……Page 141
3 The space of non-decreasing continuous functions defined on the closed unit interval ……Page 153
Bibliography ……Page 175
Index ……Page 183