which are clearly fulfilled under the assumptions (A.1) – (A.4) of [@Fr05].

\[StrongAssumption\] Let $S_i:[0,1]\to [0,1]$, $i=1,\dots,M$, denote a family of affine contractions fulfilling the open set condition (OSC), that is, $S_i((0,1)) \cap S_j((0,1))= {\varnothing}$, for all $j \neq i$, and let $\nu$ denote the associated self-similar measure with probability weight vector $(p_{1},\ldots ,p_{M}) \in (0,1)^M$ uniquely determined by $$\begin{aligned}
\nu(A)=\sum_{i=1}^{M} p_i \nu(S_i^{-1}(A))\end{aligned}$$ for $A \in \mathfrak{B}([0,1])$. For fixed $(\sigma_{1},\ldots ,\sigma_{M})\in (0,1)^{M}$ with $\sum_{i=1}^M \sigma_i \leq 1$ let $\mu$ denote an atomless probability measure with ${\operatorname{supp}}(\nu) \subseteq {\operatorname{supp}}(\mu)$ such that for all $A \in \mathfrak{B}([0,1])$, $i=1,\dots ,M$, we have $\mu(S_i(A))=\sigma_i \mu(A)$.

\[Freiberg\] For the decreasing sequence of eigenvalues $( \lambda_n )_{n \in \mathbb{N}}$ of $\Delta_{\nu,\mu}$ with domain $ \mathcal{D}^{w}_{\gamma}(\Delta_{\nu ,\mu})$, $\gamma\in [0,\pi/2]^{2}$, we have $$\begin{aligned}
  -\lambda_n\asymp n^{1/ u} \quad \text{and} \quad N_{\Delta_{\nu,\mu}}(x)\asymp x^{ u},\end{aligned}$$ where $ u\in (0,1)$ denotes the unique number with $\sum_{i=1}^M ( \sigma_i p_i )^{ u}=1.$

We have $\sigma_i F_{\mu}(x)=\sigma_i \mu([0,x])= \mu(S_i( [0,x] ) )=F_{\mu}(S_i(x))- F_{\mu}(S_i(0))$, for all $x \in [0,1]$, $i \in \{1,\dots,M\}$. Setting ${\widetilde}{S}_i(x)\coloneqq\sigma_i x+F_{\mu}(S_i(0))$ we obtain $F_{\mu} \circ S_i= {\widetilde}{S}_i \circ F_{\mu}$, and hence, for all $A \in \mathfrak{B}([0,1])$, $$\begin{aligned}
\sum_{i=1}^M p_i\nu( F_{\mu}^{-1} \circ {\widetilde}{S}_i^{-1}(A)) 
&=\sum_{i=1}^M p_i\nu(  S_i^{-1} \circ F_{\mu}^{-1}(A))
=\nu ( F_{\mu}^{-1}(A) ). 