\end{aligned}$$ This shows that $ \nu \circ F_{\mu}^{-1}$ is the unique self similar measure with contractions $( {\widetilde}{S}_i )_{i=1,\dots N}$ and weights $(p_i)_{i=1,...,M}$. Further, $$\begin{aligned}
{\widetilde}{S}_i( [0,1] )=[F_{\mu}(S_i(0)),\sigma_i+F_{\mu}(S_i(0))]=[F_{\mu}(S_i(0)),F_{\mu}(S_i(1))],\end{aligned}$$ which shows that the OSC is satisfied for $( {\widetilde}{S}_i \colon i=1,\dots ,M )$ given that $( {S}_i \colon i=1,\dots ,M )$ satisfies the OSC. Therefore, we can apply the classical result from Fujita [@Fu87] for spectral dimension of $\Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda} $ with domain ${\mathcal{D}^{w}_{\gamma} (\Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda} )}$. Combining this with completes the proof.

Let $S_i(x) \coloneqq s_ix+b_i$ for $x \in [0,1]$ with $s_i,b_i \geq 0$ and $s_i+b_i \leq 1$ for $i=1,\dots,M$ and assume the OSC. For $N\leq M$ let $\nu$ denote the self-similar measure with respect to $(S_{i}\colon i=1,\ldots ,N)$ and probability weight vector $(p_{1},\ldots , p_{N})\in (0,1)^{N}$ and support $L=\bigcup_{i=1}^N S_i(L)$, and $\mu$ the self-similar measure with respect to $(S_{i}\colon i=1,\ldots ,M)$ and probability weight vector $(\sigma_{1},\ldots , \sigma_{M})\in (0,1)^{M}$ and support $K=\bigcup_{i=1}^M S_i(K)$. Then we have $L \subset K$ and $$\begin{aligned}
\nu \circ F_{\mu}^{-1}(A)= \sum_{i=1}^N p_i \nu(S_i^{-1}\circ F_{\mu}^{-1}(A)).\end{aligned}$$ If for $\delta_{K}\coloneqq \dim_{H}(K)$ and $\delta_{L}\coloneqq \dim_{H}(L)$ we chose $\sigma_{m}\coloneqq s_{m}^{\delta_{K}}$, $m =1,\ldots, M$, and $p_{\ell}\coloneqq s_{\ell}^{\delta_{L}}$, $\ell =1,\ldots, L$, then we find the well-known 