Geometry of sets and measures in Euclidean spaces : fractals and rectifiability (Cambridge studies in advanced mathematics, 44) paper
Fractals and Rectifiability
Cambridge Studies in Advanced Mathematics Series, Vol. 44

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.