Geometry of Sets and Measures in Euclidean Spaces
Fractals and Rectifiability

This book studies the geometric properties of general sets and measures in euclidean space.

Pertti Mattila (Author)

9780521655958, Cambridge University Press

Paperback, published 25 February 1999

356 pages
22.6 x 15.2 x 2.3 cm, 0.52 kg

"Provides a unified theory for the study of the topic and develops the main tools used in its study including theorems, Hausdorff measures, and their relations to Riesz capacities and Fourier transforms." Book News, Inc.

Now in paperback, the main theme of this book is the study of geometric properties of general sets and measures in euclidean spaces. Applications of this theory include fractal-type objects such as strange attractors for dynamical systems and those fractals used as models in the sciences. The author provides a firm and unified foundation and develops all the necessary main tools, such as covering theorems, Hausdorff measures and their relations to Riesz capacities and Fourier transforms. The last third of the book is devoted to the Beisovich-Federer theory of rectifiable sets, which form in a sense the largest class of subsets of euclidean space posessing many of the properties of smooth surfaces. These sets have wide application including the higher-dimensional calculus of variations. Their relations to complex analysis and singular integrals are also studied. Essentially self-contained, this book is suitable for graduate students and researchers in mathematics.

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 othogonal projections
19. Rectifiability and analytic capacity in the complex plane
20. Rectifiability and singular intervals
References
List of notation
Index of terminology.

Subject Areas: Calculus & mathematical analysis [PBK]