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The measure theory of random fractals. (English) Zbl 0622.60021
This is a very useful survey article on measure and dimension properties of various fractals related to some stochastic processes. The basic process is the 1-parameter Brownian motion, but also its generalizations both in the 1-parameter case (Lévy processes) and the multiparameter case (Gaussian fields) are discussed. The fractals include e.g. the range, graph, level sets and sets of fixed multiplicity. Both Hausdorff and packing measure and dimension are used. The bibliography contains 100 references.
There are somewhat confusing errors in the inequalities (8) and (9) in § 4; namely, \(\mu\) (E) and \(\phi\)-m(E) in (8), and \(\mu\) (E) and \(\phi\)-p(E) in (9), have changed places. Thus e.g. (8) should read: \[ c_ 1\{\inf_{x\in E}\bar D_{\phi}(\mu,x)\}\phi -m(E)\leq \mu (E)\leq c_ 2\{\sup_{x\in E}\bar D_{\phi}(\mu,x)\}\phi -m(E). \] These corrections should be taken into account elsewhere in § 4.
Reviewer: P.Mattila

MSC:
60D05 Geometric probability and stochastic geometry
28A75 Length, area, volume, other geometric measure theory
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