Conditions on the continuity of the Hausdorff measure

Abstract: Let $b_k$ be strictly decreasing sequence of real numbers such that $b_0 = 1$ and $f_k$ be decreasing, linear functions such that $f_k(b_k) = 1$ and $f_k(b_{k-1}) = 0$, $k = 1, 2, \dots$. We define iterated function system (IFS) $S_n$ by limiting the collection of functions $f_k$ to first n, meaning $S_n = {f_k }{k=1}^n$. Let $J_n$ denote the limit set of $S_n$. We show that if $S_n$ fulfills the following two conditions: (1)~$\lim\limits{n \to \infty} \left(1-h_n\right) \ln{n} = 0 $ where $h_n$ is the Hausdorff dimension of $J_n$, and (2)~$\sup \limits_{k\in \mathbb{N}} \left {\frac{b_k-b_{k+1}}{b_{k+1}} \right } < \infty $, then $\lim\limits_{n\to \infty} H_{h_n}(J_n) = 1 = H_1(J)$, where $h_n$ is the Hausdorff dimension of $J_n$ and $H_{h_n}$ is the corresponding Hausdorff measure. We also show examples of families of IFSes fulfilling those properties.