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Statistically self-similar sets and random invariant measures. (Chinese. English summary) Zbl 1009.60043
Summary: Let $$X$$ be a complete separable metric space and $$K(\omega)$$ be the statistically self-similar sets defined by S. Graf [Probab. Theory Relat. Fields 74, 357-392 (1987; Zbl 0591.60005)]. We construct a series of random invariant measures $${\mathcal U}^*_n$$ $$(n\geq 1)$$, which are generalizations of invariant measures studied by J. E. Hutchinson [Indiana Univ. Math. J. 30, 713-747 (1981; Zbl 0598.28011)] and prove that there exists a probability measure $${\mathcal U}^*$$ with $$\text{supp }{\mathcal U}^*= K(\omega)$$ such that $${\mathcal U}^*_n\to{\mathcal U}^*$$ (weakly); finally, we obtain some local properties of $${\mathcal U}^*_n$$.

##### MSC:
 60G57 Random measures 28A80 Fractals