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Rectifiable sets, densities and tangent measures. (English) Zbl 1183.28006
Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-044-9/pbk). vi, 126 p. (2008).
The aim of this text-book is to give a self-contained approach to some deep properties of Borel measures in $$\mathbb R^ n$$. The main focus is on a well-known density property, proved by D. Preiss [Ann. Math. (2) 125, 537–643 (1987; Zbl 0627.28008)], where deep ideas and hard techniques are involved.
The present paper provides a simple and short presentation of the subject (which has already proven useful in many contexts), enjoyable for non experts. The clear style and many pictures help the understanding of the not easy matter. The result is very well framed within the general theory. An interesting list of open problems is also proposed.

##### MSC:
 28A75 Length, area, volume, other geometric measure theory 26B15 Integration of real functions of several variables: length, area, volume 49Q15 Geometric measure and integration theory, integral and normal currents in optimization 49Q20 Variational problems in a geometric measure-theoretic setting
##### MathOverflow Questions:
Preiss’ theorem on Riemannian manifolds
##### Keywords:
Borel measure; density; rectifiable sets; tangent measure
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