Conditions on the continuity of the Hausdorff measure

Let bₖ be strictly decreasing sequence of real numbers such that b₀ = 1 and fₖ be decreasing, linear functions such that fₖ(bₖ) = 1 and fₖ(bₖ₋₁) = 0, k = 1, 2,. We define iterated function system (IFS) Sₙ by limiting the collection of functions fₖ to first n, meaning Sₙ = ₖ \ₖ₌₁ⁿ. Let Jₙ denote the limit set of Sₙ. We show that if Sₙ fulfills the following two conditions: (1)~limn → ∞ (1-hₙ) lnn = 0 where hₙ is the Hausdorff dimension of Jₙ, and (2)~k∈ N ₖ-bₖ₊₁bₖ₊₁ \ < ∞, then limn→ ∞ Hhₙ(Jₙ) = 1 = H₁(J), where hₙ is the Hausdorff dimension of Jₙ and Hhₙ is the corresponding Hausdorff measure. We also show examples of families of IFSes fulfilling those properties.