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 (b₊-₁) = 0, k = 1, 2,. We define iterated function system (IFS) Sₙ by limiting the collection of functions fₖ to first n, meaning Sₙ = \fₖ \₊=₁ⁿ. Let Jₙ denote the limit set of Sₙ. We show that if Sₙ fulfills the following two conditions: (1) ~₍ (1-hₙ) n = 0 where hₙ is the Hausdorff dimension of Jₙ, and (2) ~ ₊ ₍ \bₖ-b₊+₁{b₊+₁ \} <, then ₍ H₇䂸 (Jₙ) = 1 = H₁ (J), where hₙ is the Hausdorff dimension of Jₙ and H₇䂸 is the corresponding Hausdorff measure. We also show examples of families of IFSes fulfilling those properties.