## Hausdorff MeasuresWhen 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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### Contents

MEASURES IN ABSTRACT TOPOLOGICAL AND METRIC SPACES | 1 |

3 Measures in topological spaces | 22 |

4 Measures in metric spaces | 26 |

5 Lebesgue measure in 𝑛dimensional Euclidean space | 40 |

6 Metric measures in topological spaces | 43 |

7 The Souslin operation | 44 |

HAUSDORFF MEASURES | 50 |

2 Mappings special Hausdorff measures surface areas | 53 |

6 The increasing sets lemma and its consequences | 90 |

7 The existence of comparable net measures and their properties | 101 |

8 Sets of non𝝈finite measure | 123 |

APPLICATIONS OF HAUSDORFF MEASURES | 128 |

2 Sets of real numbers defined in terms of their expansions into continued fractions | 135 |

3 The space of nondecreasing continuous functions defined on the closed unit interval | 147 |

BIBLIOGRAPHY | 169 |

APPENDIX A DIMENSION PRINTS | 177 |

### Common terms and phrases

A. S. Besicovitch Borel sets cartesian product choose a sequence class of sets closed set closed sub-intervals compact metric space compact set compact subset complete separable metric constructed by Method contains continued fraction converges Corollary countable set countable union cover cr-field cubes decreasing sequence defined definition denote dimension print Euclidean space example fi-measurable fi(E fih(E finite measure fractal function F geometric given Hausdorff dimension Hausdorff measures Hence increasing sets lemma inductively integer intervals Lebesgue measure Math measurable sets measure constructed measure on Q metric measure metric space multifractal Nn+1 obtain open sets positive integer positive number pre-measure Proc properties prove real numbers rectangle regular measure result separable metric space sequence of sets set of points sets of finite sets of Q space Q subset of finite Suppose supremum t-measurable tft-measure theorem topological space uncountable uncountable set Write zero