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Geometry of Sets and Measures in Euclidean Spaces

Geometry of Sets and Measures in Euclidean Spaces
Fractals and Rectifiability

Part of Cambridge Studies in Advanced Mathematics

  • Date Published: February 1999
  • availability: Available
  • format: Paperback
  • isbn: 9780521655958

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About the Authors
  • 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.

    • Geometric measure theory is a subject now in vogue
    • Author is an authority in the field
    • Explains the analytical mathematics behind fractals
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    Product details

    • Date Published: February 1999
    • format: Paperback
    • isbn: 9780521655958
    • length: 356 pages
    • dimensions: 226 x 152 x 23 mm
    • weight: 0.52kg
    • availability: Available
  • Table of 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 othogonal projections
    19. Rectifiability and analytic capacity in the complex plane
    20. Rectifiability and singular intervals
    References
    List of notation
    Index of terminology.

  • Author

    Pertti Mattila, University of Jyväskylä, Finland

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