The measure functions of random Cantor set and fractals determined by subordinators
A random Cantor set \(K\) on the probability space \(([0, 1], {\mathfrak B}([0, 1]), \lambda)^{\mathbb{N}}\), \(\lambda\) Lebesgue measure, is defined as follows. For \(\omega\in [0, 1]^{\mathbb{N}}\) and a convergent sequence \((a_n)_{n\in \mathbb{N}}\) such that \(a_n> 0\) for all \(n\) and \(\sum_n a_n= s\) let \[ Q_n(\omega)= \{m\mid \omega_m< \omega_n\} \text{ for all } n\in \mathbb{N},\;t_n= \sum_{s\in Q_n} a_s\text{ and } J_n= (t_n(\omega), t_n(\omega)+ a_n). \] Finally, put \[ K(\omega)= [0, s]\backslash \bigcup^\infty_{n= 1} J_n(\omega). \] The author presents results without proof concerning the exact Hausdorff as well as packing measure function of \(K(\omega)\), where \((a_n)_{n\in \mathbb{N}}\) is specified to the sequence \(\left\{{1\over 3}, {1\over 9}, {1\over 9}, {1\over 27}, {1\over 27}, {1\over 27}, {1\over 27},\dots\right\}\), and certain random product sets arising from subordinators.
scientific article; zbMATH DE number 783498