Defining fractals in a probability space
Starting with a fixed stochastic process \(X_ 1,X_ 2,\dots\) taking values in a countable set \(S\), the cylinder sets of order \(n\) consist of points \(w\in \Omega\), where \(X_ i(w)= S_ i\), \(1\leq i\leq n\). The class of all cylinder sets can be used for covering a set \(A\subset \Omega\), or for packing \(A\). This leads to a definition of Hausdorff measure or packing measure with respect to a monotone function \(\psi\). In turn this leads to three dimension indices, \(0\leq \dim A\leq\text{Dim }A\leq \Delta(A)\leq 1\), for any set \(A\subset\Omega\). The condition \(\dim A=\text{Dim }A\) is a possible condition defining a fractal set \(A\); as an example, exceptional sets for a simple random walk are shown to be fractal.
scientific article; zbMATH DE number 599244