Doubling measures on uniform Cantor sets

Abstract

We obtain a complete description for a probability measure to be doubling on an arbitrarily given uniform Cantor set. The question of which doubling measures on such a Cantor set can be extended to a doubling measure on [0; 1] is also considered

On the Doubling Condition in the Infinite-Dimensional Setting

“…This can be done through the bijection given by . Then, the usual Cantor measure on is doubling, since for all and , one has (see also [16]).…”

“…Indeed, it suffices to identify {0, 1 2 } ω with the classical Cantor set C ⊂ [0, 1] with the metric ρ C obtained by restricting ρ R to C × C. This can be done through the bijection π : {0, 1 2 } ω → C given by π(x) ∞ n=1 4x n /3 n . Then, the usual Cantor measure μ C on C is doubling, since for all x ∈ C and n ∈ N, one has μ C (B ρ C (3 −n )) = 2 −n (see also [16]).…”

On the Asymptotic Uniformity of the Quantization Error for Moran Measures on ℝ1

“…By Proposition 1.5 of [29], one can see that a doubling measure ν carried by F needs not to be an Ahlfors-David measure, although it is well known that under the OSC, F is an s-set and the normalized s-dimensional Hausdorff measure H s (·|E) is an s-dimensional Ahlfors-David measure. One may also see [27] for characterizations for the doubling measures carried by some Moran sets.…”