Generalized random ergodic theorems and Hausdorff-measures of random fractals (Q914232)

For the random version of Hutchinson's self-similar sets introduced by \textit{K. J. Falconer} [Math. Proc. Camb. Philos. Soc. 100, 559-582 (1986; Zbl 0623.60020)], \textit{S. Graf} [Probab. Theory Relat. Fields 74, 357-392 (1987; Zbl 0591.60005)], and \textit{R. D. Mauldin} and \textit{S. C. Williams} [Trans. Am. Math. Soc. 295, 325-346 (1986; Zbl 0625.54047)], the Hausdorff dimension is a.s. a constant which may be computed by means of the contraction ratios of the underlying similarities. Moreover, the corresponding random Hausdorff measure has finite expectation. The author proves here that the measure is a.s. constant. Thereby he uses an extension of Markov ergodicity results of \textit{S. Kakutani} [Proc. Berkeley Sympos. Math. Statist. Probability, California July 31-August 12, 1950, 247-261 (1951; Zbl 0044.339)] to generalized random dynamical systems given in the paper.