Book contents
- Frontmatter
- Contents
- Acknowledgements
- Basic notation
- Introduction
- 1 General measure theory
- 2 Covering and differentiation
- 3 Invariant measures
- 4 Hausdorff measures and dimension
- 5 Other measures and dimensions
- 6 Density theorems for Hausdorff and packing measures
- 7 Lipschitz maps
- 8 Energies, capacities and subsets of finite measure
- 9 Orthogonal projections
- 10 Intersections with planes
- 11 Local structure of s-dimensional sets and measures
- 12 The Fourier transform and its applications
- 13 Intersections of general sets
- 14 Tangent measures and densities
- 15 Rectifiable sets and approximate tangent planes
- 16 Rectifiability, weak linear approximation and tangent measures
- 17 Rectifiability and densities
- 18 Rectifiability and orthogonal projections
- 19 Rectifiability and analytic capacity in the complex plane
- 20 Rectifiability and singular integrals
- References
- List of notation
- Index of terminology
Introduction
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Acknowledgements
- Basic notation
- Introduction
- 1 General measure theory
- 2 Covering and differentiation
- 3 Invariant measures
- 4 Hausdorff measures and dimension
- 5 Other measures and dimensions
- 6 Density theorems for Hausdorff and packing measures
- 7 Lipschitz maps
- 8 Energies, capacities and subsets of finite measure
- 9 Orthogonal projections
- 10 Intersections with planes
- 11 Local structure of s-dimensional sets and measures
- 12 The Fourier transform and its applications
- 13 Intersections of general sets
- 14 Tangent measures and densities
- 15 Rectifiable sets and approximate tangent planes
- 16 Rectifiability, weak linear approximation and tangent measures
- 17 Rectifiability and densities
- 18 Rectifiability and orthogonal projections
- 19 Rectifiability and analytic capacity in the complex plane
- 20 Rectifiability and singular integrals
- References
- List of notation
- Index of terminology
Summary
This is a book on geometric measure theory. The main theme is the study of the geometric structure of general Borel sets and Borel measures in the euclidean n-space Rn. There will be emphasis on “small irregular” sets having Lebesgue measure zero but being quite different from smooth curves and surfaces. Examples are Cantor-type sets, nonrectifiable curves having tangent nowhere, etc., in short, sets to which the general descriptive term fractal applies. An abundance of such sets comes from dynamical systems: Julia-sets for rational functions of one complex variable, etc. Very general curve – and surface-like objects are also studied extensively. These are rectifiable sets and measures. They include smooth curves and surfaces and share many of their geometric properties when interpreted in a measure-theoretic sense. They form an optimal class possessing such properties.
Many of the basic ideas developed here originate in the pioneering work done by Besicovitch [1], [4] and [5], by Federer [1], by Marstrand [1] and by Preiss [4]. Besicovitch laid down the foundations of geometric measure theory by describing to an amazing extent the structure of the subsets of the plane having finite one-dimensional Hausdorff measure (i.e. length). Federer extended Besicovitch's work to m-dimensional subsets of Rn, m being an integer, and Marstrand analysed general fractals in the plane whose Hausdorff dimension need not be an integer. Preiss solved one of the most long-standing fundamental open problems, introducing and using effectively tangent measures.
Good introductory texts to the mathematical theory of fractals are the books of Edgar [1] and of Falconer [4], [16]. Closest to this text is Falconer [4].
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- Chapter
- Information
- Geometry of Sets and Measures in Euclidean SpacesFractals and Rectifiability, pp. 1 - 6Publisher: Cambridge University PressPrint publication year: 1995