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We weaken the open set condition and define a finite intersectionproperty in the construction of the random recursive sets. We prove that this larger class of random sets are fractals in the sense of Taylor, and give conditions when these sets have positive and finite Hausdorff measures, which in certain extent generalize some of the known results, about random recursive fractals

The Hausdorff dimension and exact Hausdorff measure of random recursive sets with overlapping