Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. (English) Zbl 0819.28004

Cambridge Studies in Advanced Mathematics. 44. Cambridge: Univ. Press. xii, 343 p. (1995).
This well-written book begins by describing the general measure theory as well as the required tools for geometric measure theory mainly based on Hausdorff measures in the first seven chapters. Later on, the author gives a complete overview on newer tools amongst them packing measures and Preiss’ tangent measures. After an interplay highlighting the role of analytical methods such as Fourier transforms, rectifiability and its interaction with other geometric properties become the main topics in the last third of this book. These chapters reflect the recent advances in the geometric measure theory from the Besicovitch-Federer point of view, for example Preiss’ theorem, but also own results of the author in this field. The presentation of each chapter shows that the author is a very experienced lecturer directing reader’s attention to the key point and avoiding superfluous technicalities. This book is useful for graduate students as well as researchers in geometric measure theory.


28A75 Length, area, volume, other geometric measure theory
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration
28A78 Hausdorff and packing measures
28A80 Fractals