O'Neil, Toby Christopher; (1995) A local version of the projection theorem and other results in geometric measure theory. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

In this thesis we investigate how knowledge of the local behaviour of a Borel measure on Rn enables us to deduce information about its global behaviour. The main concept we use for this is that of tangent measures as introduced by Preiss. In order to illustrate the limitations of tangent measures we first construct a Borel measure μ on Rn such that for μ-a.e. x, all non-zero, locally finite Borel measures on Rn are tangent measures of μμ at x. Furthermore we show that the set of measures for which this fails to be true is of first category in the space of Borel measures on Rn. The main result of the thesis is the following: Suppose that 1 < m < n are integers and μ is a Borel measure on Rn such that for μ-a.e.x, 1. The upper and lower m-densities of μ at x are positive and finite. 2. If v is a tangent measure of μ at x then for all V E G(n, m) the orthogonal projection of the support of v onto V is a convex set. Then μ is m-rectifiable. By considering a measure derived from a variation of an example given by Dickinson, we are able to illustrate the necessity of a condition such as (2) in our main theorem. Moreover this measure has its average density equal to its upper density and possesses a unique tangent measure distribution almost everywhere. Our final example is based upon one given by Besicovitch. We show that there is a Borel measure μ with positive and finite upper and lower 1-density almost everywhere and with average 1-density existing almost everywhere but with non-unique tangent measure distributions.

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