New concepts of Lebesgue measure on are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on for α = (1,1,...).
We investigate the definition and measurability questions of random fractals with infinite branching, and find, under certain conditions, a formula for the upper and lower Minkowski dimensions. For the case of a random self-similar set we obtain the packing dimension.
In this paper, we first give several properties with respect to subgroups of self-similar groups in the sense of iterated function system (IFS). We then prove that some subgroups of -adic numbers are strong self-similar in the sense of IFS.