Lorent, Andrew; (1999) Rectifiability results for l3∞. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

As a first step to generalising Rectifiability and Density Results, Radon measures with density properties with respect to the cube are studied. For reasons of isometric immersion of metric spaces into l∞ and the extremal nature of l3∞ among finite dimensional normed vector spaces, the question of rectifiability of such measures is the simplest unknown case of any generalisation. It is proved that locally 2-uniform measures in l3∞ have rectifiable subsets in all neighbourhoods of all points of their support. By a well known theorem on tangent measures an immediate Corollary to this is that measures with positive finite 2-density almost everywhere have weak tangents.

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