# Rectifiable Sets, Densities and Tangent Measures

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*Camillo De Lellis*

A publication of the European Mathematical Society

The characterization of rectifiable sets through the existence of
densities is a pearl of geometric measure theory. The difficult proof, due to
Preiss, relies on many beautiful and deep ideas and novel techniques. Some of
them have already proven useful in other contexts, whereas others have not yet
been exploited. These notes give a simple and short presentation of the former
and provide some perspective of the latter.

This text emerged from a course on rectifiability given at the University of
Zürich. It is addressed both to researchers and students; the only
prerequisite is a solid knowledge in standard measure theory. The first four
chapters give an introduction to rectifiable sets and measures in Euclidean
spaces, covering classical topics such as the area formula, the theorem of
Marstrand and the most elementary rectifiability criterions. The fifth chapter
is dedicated to a subtle rectifiability criterion due to Marstrand and
generalized by Mattila, and the last three focus on Preiss' result. The aim is
to provide a self-contained reference for anyone interested in an overview of
this fascinating topic.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in analysis.