Moerters, Peter; (1995) Tangent measure distributions and the geometry of measures. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

In this thesis we investigate the geometry of measures in Euclidean spaces by means of their average densities, average tangent measures and tangent measure distributions. These notions were recently introduced into geometric measure theory by Bedford and Fisher, Bandt, Graf and others, as tools for the study of non-rectifiable measures. Our main result yields a connection between tangent measure distributions of measures on the line and Palm distributions: Let [alpha] be a measure on the line with positive and finite [alpha]-densities almost everywhere. Then at almost all points all tangent measure distributions are Palm distributions. Therefore the tangent measure distributions define a-self similar random measures in the axiomatic sense of U. Zahle. This result enables us to give a complete description of the one-sided average [alpha]-densities of the measure in terms of its lower and upper circular average [alpha]-densities. It also enables us to give an example of a measure with positive and finite [alpha]-densities which has unique average tangent measures but non-unique tangent measure distributions almost everywhere. If [mu] is a measure on n-dimensional Euclidean space with positive and finite [alpha]-densities almost everywhere we show that at almost all points the unique tangent measure distribution, if it exists, is a Palm distribution. We illustrate the limitations of tangent measure distributions by means of an example of a non-zero measure that has no non-trivial tangent measure distributions almost everywhere. Such measures can be studied by means of normalized tangent measure distributions and we prove an existence and a shift-invariance result for these distributions.

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