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On some notions of tangent space to a measure

Published online by Cambridge University Press:  14 November 2011

Ilaria Fragalà  and
Carlo Mantegazza
Ilaria Fragalà
Affiliation:
Università di Pisa, Dipartimento di Matematica, Via Buonarroti, 2, 56127 Pisa, Italy, ([email protected])
Carlo Mantegazza
Affiliation:
Scuola Normale Superiore di Pisa, Classe di Scienze, Piazza dei Cavalieri, 7, 56126 Pisa, Italy, ([email protected])
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Abstract

We consider some definitions of tangent space to a Radon measure μ on ℝn that have been given in the literature. In particular, we focus our attention on a recent distributional notion of tangent vector field to a measure and we compare it to other definitions coming from ‘geometric measure theory’, based on the idea of blow-up. After showing some classes of examples, we prove an estimate from above for the dimension of the tangent spaces and a rectifiability theorem which also includes the case of measures supported on sets of variable dimension.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 2 , 1999 , pp. 331 - 342
Copyright
Copyright © Royal Society of Edinburgh 1999

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