Acknowledgements 

xi  
Basic notation 

xii  
Introduction 

1  (6) 


7  (16) 


7  (1) 


8  (5) 


13  (2) 


15  (3) 


18  (1) 


19  (3) 


22  (1) 

Covering and differentiation 


23  (21) 


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) 

HardyLittlewood maximal function 


40  (2) 

Measures in infinite dimensional spaces 


42  (1) 


43  (1) 


44  (10) 


44  (1) 

Uniformly distributed measures 


45  (1) 


46  (2) 

The Grassmannian of mplanes 


48  (4) 


52  (1) 


53  (1) 


53  (1) 

Hausdorff measures and dimension 


54  (21) 

Caratheodory's construction 


54  (1) 


55  (3) 


58  (1) 

Generalized Hausdorff measures 


59  (1) 


60  (5) 

Selfsimilar and related sets 


65  (4) 

Limit sets of Mobius groups 


69  (2) 

Dynamical systems and Julia sets 


71  (1) 


72  (1) 


73  (2) 

Other measures and dimensions 


75  (14) 


75  (1) 


76  (1) 


76  (5) 

Packing dimensions and measures 


81  (5) 

Integralgeometric measures 


86  (2) 


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) 


99  (1) 


100  (9) 

Extension of Lipschitz maps 


100  (1) 

Differentiability of Lipschitz maps 


100  (3) 


103  (1) 

Hausdorff measures of level sets 


104  (1) 

The lower density of Lipschitz images 


105  (1) 

Remarks on Lipschitz maps 


106  (1) 


107  (2) 

Energies, capacities and subsets of finite measure 


109  (17) 


109  (1) 

Capacities and Hausdorff measures 


110  (2) 


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) 


124  (2) 


126  (13) 

Lipschitz maps and capacities 


126  (1) 

Orthogonal projections, capacities and Hausdorff dimension 


127  (7) 

Selfsimilar sets with overlap 


134  (2) 


136  (2) 


138  (1) 

Intersections with planes 


139  (7) 

Slicing measures with planes 


139  (3) 

Plane sections, capacities and Hausdorff measures 


142  (3) 


145  (1) 

Local structure of sdimensional sets and measures 


146  (13) 

Distribution of measures with finite energy 


146  (6) 


152  (4) 

Porosity and Hausdorff dimension 


156  (2) 


158  (1) 

The Fourier transform and its applications 


159  (12) 


159  (3) 

The Fourier transform and energies 


162  (3) 


165  (1) 


166  (2) 

Fourier dimension and Salem sets 


168  (1) 


169  (2) 

Intersections of general sets 


171  (13) 

Intersection measures and energies 


171  (6) 

Hausdorff dimension and capacities of intersections 


177  (3) 


180  (2) 


182  (2) 

Tangent measures and densities 


184  (18) 


184  (2) 

Preliminary results on tangent measures 


186  (3) 

Densities and tangent measures 


189  (2) 


191  (1) 


192  (2) 


194  (2) 

Tangent measures to tangent measures are tangent measures 


196  (2) 


198  (2) 


200  (1) 


200  (2) 

Rectifiable sets and approximate tangent planes 


202  (18) 


202  (1) 


203  (2) 

Linear approximation properties 


205  (3) 

Rectifiability and measures in cones 


208  (4) 

Approximate tangent planes 


212  (2) 

Remarks on rectifiability 


214  (1) 


215  (3) 


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) 


230  (1) 

Rectifiability and densities 


231  (19) 

Structure of muniform measures 


231  (9) 

Rectifiability and density one 


240  (1) 


241  (6) 

Rectifiability and packing measures 


247  (1) 


247  (2) 


249  (1) 

Rectifiability and orthogonal projections 


250  (15) 

BesicovitchFederer projection theorem 


250  (8) 


258  (2) 


260  (4) 


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) 


276  (3) 


279  (2) 

Rectifiability and singular integrals 


281  (24) 


281  (2) 


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) 

Lpboundedness and weak (1,1) inequalities 


289  (3) 

A duality method for weak (1,1) 


292  (3) 

A smoothing of singular integral operators 


295  (3) 


298  (1) 


299  (2) 

A weak (1, 1) inequality for complex measures 


301  (1) 

Rectifiability implies existence of principal values 


301  (3) 


304  (1) 
References 

305  (29) 
List of notation 

334  (3) 
Index of terminology 

337  