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Geometry of Sets and Measures in Euclidean Spaces : Fractals and Rectifiability,9780521655958
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Geometry of Sets and Measures in Euclidean Spaces : Fractals and Rectifiability

by
ISBN13:

9780521655958

ISBN10:
0521655951
Format:
Paperback
Pub. Date:
3/28/1999
Publisher(s):
Cambridge University Press
List Price: $91.00

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This is the edition with a publication date of 3/28/1999.
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Summary

The focus of this book is 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 for the subject and develops all the main tools used in its study, 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 Besicovitch-Federer theory of rectifiable sets, which form in a sense the largest class of subsets of Euclidean space possessing many of the properties of smooth surfaces.

Table of Contents

Acknowledgements xi
Basic notation xii
Introduction 1(6)
General measure theory
7(16)
Some basic notation
7(1)
Measures
8(5)
Integrals
13(2)
Image measures
15(3)
Weak convergence
18(1)
Approximate identities
19(3)
Exercises
22(1)
Covering and differentiation
23(21)
A 5r-covering theorem
23(3)
Vitali's covering theorem for the Lebesgue measure
26(2)
Besicovitch's covering theorem
28(6)
Vitali's covering theorem for Radon measures
34(1)
Differentiation of measures
35(5)
Hardy--Littlewood maximal function
40(2)
Measures in infinite dimensional spaces
42(1)
Exercises
43(1)
Invariant measures
44(10)
Haar measure
44(1)
Uniformly distributed measures
45(1)
The orthogonal group
46(2)
The Grassmannian of m-planes
48(4)
The isometry group
52(1)
The affine subspaces
53(1)
Exercises
53(1)
Hausdorff measures and dimension
54(21)
Caratheodory's construction
54(1)
Hausdorff measures
55(3)
Hausdorff dimension
58(1)
Generalized Hausdorff measures
59(1)
Cantor sets
60(5)
Self-similar and related sets
65(4)
Limit sets of Mobius groups
69(2)
Dynamical systems and Julia sets
71(1)
Harmonic measure
72(1)
Exercises
73(2)
Other measures and dimensions
75(14)
Spherical measures
75(1)
Net measures
76(1)
Minkowski dimensions
76(5)
Packing dimensions and measures
81(5)
Integralgeometric measures
86(2)
Exercises
88(1)
Density theorems for Hausdorff and packing measures
89(11)
Density estimates for Hausdorff measures
89(3)
A density theorem for spherical measures
92(2)
Densities of Radon measures
94(1)
Density theorems for packing measures
95(3)
Remarks related to densities
98(1)
Exercises
99(1)
Lipschitz maps
100(9)
Extension of Lipschitz maps
100(1)
Differentiability of Lipschitz maps
100(3)
A Sard-type theorem
103(1)
Hausdorff measures of level sets
104(1)
The lower density of Lipschitz images
105(1)
Remarks on Lipschitz maps
106(1)
Exercises
107(2)
Energies, capacities and subsets of finite measure
109(17)
Energies
109(1)
Capacities and Hausdorff measures
110(2)
Frostman's lemma in Rn
112(3)
Dimensions of product sets
115(2)
Weighted Hausdorff measures
117(3)
Frostman's lemma in compact metric spaces
120(1)
Existence of subsets with finite Hausdorff measure
121(3)
Exercises
124(2)
Orthogonal projections
126(13)
Lipschitz maps and capacities
126(1)
Orthogonal projections, capacities and Hausdorff dimension
127(7)
Self-similar sets with overlap
134(2)
Brownian motion
136(2)
Exercises
138(1)
Intersections with planes
139(7)
Slicing measures with planes
139(3)
Plane sections, capacities and Hausdorff measures
142(3)
Exercises
145(1)
Local structure of s-dimensional sets and measures
146(13)
Distribution of measures with finite energy
146(6)
Conical densities
152(4)
Porosity and Hausdorff dimension
156(2)
Exercises
158(1)
The Fourier transform and its applications
159(12)
Basic formulas
159(3)
The Fourier transform and energies
162(3)
Distance sets
165(1)
Borel subrings of R
166(2)
Fourier dimension and Salem sets
168(1)
Exercises
169(2)
Intersections of general sets
171(13)
Intersection measures and energies
171(6)
Hausdorff dimension and capacities of intersections
177(3)
Examples and remarks
180(2)
Exercises
182(2)
Tangent measures and densities
184(18)
Definitions and examples
184(2)
Preliminary results on tangent measures
186(3)
Densities and tangent measures
189(2)
s-uniform measures
191(1)
Marstrand's theorem
192(2)
A metric on measures
194(2)
Tangent measures to tangent measures are tangent measures
196(2)
Proof of Theorem 11.11
198(2)
Remarks
200(1)
Exercises
200(2)
Rectifiable sets and approximate tangent planes
202(18)
Two examples
202(1)
m-rectifiable sets
203(2)
Linear approximation properties
205(3)
Rectifiability and measures in cones
208(4)
Approximate tangent planes
212(2)
Remarks on rectifiability
214(1)
Uniform rectifiability
215(3)
Exercises
218(2)
Rectifiability, weak linear approximation and tangent measures
220(11)
A lemma on projections of purely unrectifiable sets
220(2)
Weak linear approximation, densities and projections
222(6)
Rectifiability and tangent measures
228(2)
Exercises
230(1)
Rectifiability and densities
231(19)
Structure of m-uniform measures
231(9)
Rectifiability and density one
240(1)
Preiss's theorem
241(6)
Rectifiability and packing measures
247(1)
Remarks
247(2)
Exercises
249(1)
Rectifiability and orthogonal projections
250(15)
Besicovitch-Federer projection theorem
250(8)
Remarks on projections
258(2)
Besicovitch sets
260(4)
Exercises
264(1)
Rectifiability and analytic capacity in the complex plane
265(16)
Analytic capacity and removable sets
265(2)
Analytic capacity, Riesz capacity and Hausdorff measures
267(1)
Cauchy transforms of complex measures
267(6)
Cauchy transforms and tangent measures
273(2)
Analytic capacity and rectifiability
275(1)
Various remarks
276(3)
Exercises
279(2)
Rectifiability and singular integrals
281(24)
Basic singular integrals
281(2)
Symmetric measures
283(1)
Existence of principal values and tangent measures
284(1)
Symmetric measures with density bounds
285(3)
Existence of principal values implies rectifiability
288(1)
Lp-boundedness and weak (1,1) inequalities
289(3)
A duality method for weak (1,1)
292(3)
A smoothing of singular integral operators
295(3)
Kolmogorov's inequality
298(1)
Cotlar's inequality
299(2)
A weak (1, 1) inequality for complex measures
301(1)
Rectifiability implies existence of principal values
301(3)
Exercises
304(1)
References 305(29)
List of notation 334(3)
Index of terminology 337


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