Random point fields with Markovian refinements and the geometry of fractally disordered media

We give a general construction of the probability measure for describing stochastic fractals that model fractally disordered media. For these stochastic fractals, we introduce the notion of a metrically homogeneous fractal Hausdorff–Karathéodory measure of a nonrandom type. We select a class $\mathbf F[q]$ of random point fields with Markovian refinements for which we explicitly construct the probability distribution. We prove that under rather weak conditions, the fractal dimension $D$ for random fields of this class is a self-averaging quantity and a fractal measure of a nonrandom type (the Hausdorff $D$-measure) can be defined on these fractals with probability 1.