## Geometry of Sets and Measures in Euclidean Spaces: Fractals and RectifiabilityThe main theme of this book is the study of geometric properties of general sets and measures in euc lidean space. Examples to which this theory applies 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. Thus the book is essentially self-contained for graduate students in mathematics; it is primarily targetted at them and researchers.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. These sets have wide application; for example they are central in the higher-dimensional calculus of variations. Their relations to complex analysis and singular integrals are also studied. |

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### Sadržaj

1 General measure theory | 7 |

Measures | 8 |

Integrals | 13 |

Image measures | 15 |

Weak convergence | 18 |

Approximate identities | 19 |

Exercises | 22 |

2 Covering and differentiation | 23 |

Porosity and Hausdorff dimension | 156 |

Exercises | 158 |

12 The Fourier transform and its applications | 159 |

The Fourier transform and energies | 162 |

Distance sets | 165 |

Borel subrings of R | 166 |

Fourier dimension and Salem sets | 168 |

Exercises | 169 |

Vitalis covering theorem for the Lebesgue measure | 26 |

Besicovitchs covering theorem | 28 |

Vitalis covering theorem for Radon measures | 34 |

Differentiation of measures | 35 |

HardyLittlewood maximal function | 40 |

Measures in infinite dimensional spaces | 42 |

Exercises | 43 |

3 Invariant measures Haar measure | 44 |

Uniformly distributed measures | 45 |

The orthogonal group | 46 |

The Grassmannian of mplanes | 48 |

The isometry group | 52 |

The affine subspaces | 53 |

4 Hausdorff measures and dimension | 54 |

Hausdorff measures | 55 |

Hausdorff dimension | 58 |

Generalized Hausdorff measures | 59 |

Cantor sets | 60 |

Selfsimilar and related sets | 65 |

Limit sets of Möbius groups | 69 |

Dynamical systems and Julia sets | 71 |

Harmonic measure | 72 |

Exercises | 73 |

5 Other measures and dimensions | 75 |

Net measures | 76 |

Packing dimensions and measures | 81 |

Integralgeometric measures | 86 |

Exercises | 88 |

6 Density theorems for Hausdorff and packing measures | 89 |

A density theorem for spherical measures | 92 |

Densities of Radon measures | 94 |

Density theorems for packing measures | 95 |

Remarks related to densities | 98 |

Exercises | 99 |

7 Lipschitz maps | 100 |

A Sardtype theorem | 103 |

Hausdorff measures of level sets | 104 |

The lower density of Lipschitz images | 105 |

Remarks on Lipschitz maps | 106 |

Exercises | 107 |

8 Energies capacities and subsets of finite measure | 109 |

Capacities and Hausdorff measures | 110 |

Frostmans lemma in Rn | 112 |

Dimensions of product sets | 115 |

Weighted Hausdorff measures | 117 |

Frostmans lemma in compact metric spaces | 120 |

Existence of subsets with finite Hausdorff measure | 121 |

Exercises | 124 |

9 Orthogonal projections | 126 |

Orthogonal projections capacities and Hausdorff dimension | 127 |

Selfsimilar sets with overlap | 134 |

Brownian motion | 136 |

Exercises | 138 |

10 Intersections with planes | 139 |

Plane sections capacities and Hausdorff measures | 142 |

Exercises | 145 |

11 Local structure of sdimensional sets and measures | 146 |

Conical densities | 152 |

13 Intersections of general sets | 171 |

Hausdorff dimension and capacities of intersections | 177 |

Examples and remarks | 180 |

Exercises | 182 |

14 Tangent measures and densities | 184 |

Preliminary results on tangent measures | 186 |

Densities and tangent measures | 189 |

suniform measures | 191 |

Marstrands theorem | 192 |

A metric on measures | 194 |

Tangent measures to tangent measures are tangent measures | 196 |

Proof of Theorem 1111 | 198 |

Remarks | 200 |

15 Rectifiable sets and approximate tangent planes | 202 |

mrectifiable sets | 203 |

Linear approximation properties | 205 |

Rectifiability and measures in cones | 208 |

Approximate tangent planes | 212 |

Remarks on rectifiability | 214 |

Uniform rectifiability | 215 |

Exercises | 218 |

16 Rectifiability weak linear approximation and tangent measures | 220 |

Weak linear approximation densities and projections | 222 |

Rectifiability and tangent measures | 228 |

Exercises | 230 |

17 Rectifiability and densities | 231 |

Rectifiability and density one | 240 |

Preisss theorem | 241 |

Rectifiability and packing measures | 247 |

Exercises | 249 |

18 Rectifiability and orthogonal projections | 250 |

Remarks on projections | 258 |

Besicovitch sets | 260 |

Exercises | 264 |

19 Rectifiability and analytic capacity in the complex plane | 265 |

Analytic capacity Riesz capacity and Hausdorff measures | 267 |

Cauchy transforms of complex measures | 269 |

Cauchy transforms and tangent measures | 273 |

Analytic capacity and rectifiability | 275 |

Various remarks | 276 |

Exercises | 279 |

20 Rectifiability and singular integrals Basic singular integrals | 281 |

Symmetric measures | 283 |

Existence of principal values and tangent measures | 284 |

Symmetric measures with density bounds | 285 |

Existence of principal values implies rectifiability | 288 |

Lpboundedness and weak 11 inequalities | 289 |

A duality method for weak 11 | 292 |

A smoothing of singular integral operators | 295 |

Kolmogorovs inequality | 298 |

Collars inequality | 299 |

Rectifiability implies existence of principal values | 301 |

Exercises | 304 |

References | 305 |

List of notation | 334 |

337 | |

### Ostala izdanja - Prikaži sve

Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability Pertti Mattila Ograničeni pregled - 1999 |

Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability Pertti Mattila Pregled nije dostupan - 1995 |

### Uobičajeni izrazi i fraze

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