P. M orters, \Tangent Measure Distributions and the Geometry of Measures", PhD thesis, University College London, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on the line. However, in the important case of measures on the line with positive lower and nite upper densities it is possible to deduce the Palm property of tangent measure distributions from very detailed information about the structure of these measures which has been obtained in the thesis [M o1] and will appear in [M o2] 2. The main result and its applications In this section we give the full de nitions and basic properties of the main geometric measure theoretical notions describing the dimensional part of the behaviour of a Radon measure on R d about a point x 2 R d and we ....

P. Morters. Tangent Measure Distributions and the Geometry of Measures. PhD thesis, University College London, 1995.

.... 1 E ( 1;2 ) x;t t dt t d 1;2 (x) where E 2 M b , the algebra on M(IR) generated by the mappings 7 (B) for all Borel sets B B(0; b) and (1= 7b) It is easy to see that R ( 1; 1] dP 1 ( R ( 1; 1] dP 2 ( Using messy but straightforward calculations (see [15] for details) we can also see that for almost all points x 2 I the set of tangent measure distributions of at x is given by f P 1 (1 )P 2 : 2 [0; 1]g : This not only shows that the set of tangent measure distributions of at x is not a singleton but (with the help of (3) also ....

P. M orters, \Tangent Measure Distributions and the Geometry of Measures", PhD thesis, University College London, 1995. 24

....the argument in Section 3 of this paper. Do the inequalities above also imply ff rectifiability of The answer is no for the first inequality, even in the case ff = 1. The first example of this type can be found in [O N95] Let us sketch a simpler example here, more details can be found in [Mor95]. Denote by B(x; r) the closed ball of radius x centred in r. Example: We construct a sequence (I n ) of compact sets I n IR 2 as follows: Let I 0 = B(0; 1) In the first step inscribe 2 touching, closed balls of radius 1=2 with centres on the vertical diameter in I 0 . Denote the resulting ....

....observe that every small ball centred in I of radius r must contain disjoint balls B 2 B whose summed diameters exceed r=2 and may be covered by a family of balls B 2 B whose summed diameter does not exceed 2r. Hence d 1 ( x) 1=2 and d 1 ( x) 2. By a more detailed analysis, carried out in [Mor95] one finds that there is a Z (0; 1) of vanishing logarithmic density, i.e. such that lim #0 1 log(1= Z 1 1 Z (r) dr r = 0 ; with the property that lim r#0 r62Z (U(x; r) r = 2 : Hence D 1 ( x) D 1 ( x) 2 and, finally, it is not hard to find sequences r n # 0 such ....

[Article contains additional citation context not shown here]

P. M orters, Tangent Measure Distributions and the Geometry of Measures. PhD thesis, University College London, 1995.

....measure distributions of at x. Tangent measure distributions were introduced by Bandt ( Ban92] and Graf ( Gra95] originally as a tool for the investigation of self similar sets. They have also turned out to be valuable for the study of more general measures (see [Mor96] MP96] or the thesis [Mor95]) which is due to the invariance properties described in the following theorem. For every 0 we define the rescaling operator S ff : M(IR d ) M(IR d ) by S ff (E) 1= ff ) Delta (E) and for every u 2 IR d we define the shift operator T u : M(IR d ) M(IR d ) by T u (E) ....

P. M orters, Tangent Measure Distributions and the Geometry of Measures. PhD thesis, University College London, 1995.

....distributions on the set of tangent measures. We now state some general properties of tangent measure distributions for measures , with positive lower and finite upper s densities, which will be used in the following investigations. The following properties can be checked easily (for proofs see [12]) Proposition 1.1 The closure of f a;t =t s : 0 t 1g M(IR) is compact. Consequently, the sets Tan s ( a) and P( a) are nonempty and P is the unique tangent measure distribution of at a if and only if P( a) fPg. Proposition 1.2 Suppose , are equivalent measures and f = d ....

....G( u) d(u) dP ( Z Z G(T u ; Gammau) d(u) dP ( for all Borel functions G : M(IR) Theta IR Gamma [0; 1) 1) Whereas the scaling invariance 1.3(i) is easy to check, the shift invariance 1.3(ii) the so called Palm formula, is non trivial. A proof can be found in [13] or the thesis [12]. We can use these invariance properties to see what happens if we iterate the procedure of taking tangent measure distributions. Proposition 1.4 Let 2 M(IR) be a measure with positive lower and finite upper s densities almost everywhere. Then, at almost every point, every tangent measure ....

[Article contains additional citation context not shown here]

P. M orters. Tangent Measure Distributions and the Geometry of Measures. PhD thesis, University College London, 1995.

No context found.

P. M orters, \Tangent Measure Distributions and the Geometry of Measures", PhD thesis, University College London, 1995.

No context found.

P. M orters, Tangent Measure Distributions and the Geometry of Measures. PhD thesis, University College London, 1995.

No context found.

P. Morters. Tangent Measure Distributions and the Geometry of Measures. PhD thesis, University College London, 1995.

No context found.

P. M orters, Tangent Measure Distributions and the Geometry of Measures. PhD thesis, University College London, 1995.

No context found.

P. M orters, Tangent Measure Distributions and the Geometry of Measures. PhD thesis, University College London, 1995.

No context found.

P. M orters. Tangent Measure Distributions and the Geometry of Measures. PhD thesis, University College London, 1995.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC