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Lecture notes on geometric measure theory. (English) Zbl 0638.28006
Publicaciones del Departamento de Matemáticas, Universidad de Extremadura, 14. Badajoz (España): Universidad de Extremadura, Facultad de Ciencias, Departamento de Matemáticas. V, 122 p. (1986).
The notion of geometric measure is the generalization of the familiar notions of curve length, surface area and so on. Moreover, this notion is acceptable for fractals. Fractals, or the sets of fractionary dimension, arise e.g. in number theory, in probability as the level set of stable processes or as the range of increasing processes, without drift, in dynamical systems as strange attractors etc.
In the present lecture notes the author gives in a manner accessible to students a survey of the geometric structure of general subsets of \(R^ n\) in the context of geometric measure theory. This lecture notes have no much overlaps with the recent K. J. Falconer’s book [Geometry of fractal sets (1985; Zbl 0587.28004)].
At first the author recalls the basic notions of general measure theory and the covering theorems of Vitali and Besicovitch. Then the general construction of Hausdorff measure, accompanied by several examples, including the theory of self-similar sets, is presented. This construction is the basis of all further constructions. The next topics are estimation of density for Hausdorff measure, density properties of fractals, structure of sets having finite m-dimensional Hausdorff measure, relation between capacities and Hausdorff measures, their changing under orthogonal projection etc.
A full bibliography, including some recent papers on this topic, is added.
Reviewer: I.S.Molchanov

28A75 Length, area, volume, other geometric measure theory
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration