domain for the action of $N_n$ on $K$, with $n$ large enough. Hence, we have for every $\alpha\in {\mathcal{SC}_*}$: $$\begin{aligned}
\mu_{K}(U_\alpha(r))&=\frac{|\{x\in V({\mathcal F}(N_n))\mid B_r(K,x)\cong \alpha\}|}{|V({\mathcal F}(N_n))|} & \text{ cf. Example \ref{example-measures}\,(2)} \\
&= \frac{|\{x\in V(K_n)\mid B_r(K_n,xN_n)\cong \alpha\}|}{|V(K_n)|}&\\
&=\mu_{K_n}(U_\alpha(r)).\end{aligned}$$ Let $K$ be a simplicial complex with a cocompact action of a sofic group, then there exist a Benjamini-Schramm convergent sequence of simplicial complexes converging to $\mu_K$.

In the following we will present our version of Lück’s approximation Theorem [@lueck]. The proof adapts the ideas of [@abert-thom-virag] to higher dimensions, where Abért, Thom and Virág proved a similar result for graphs [@abert-thom-virag]. They showed that for sofic measures of uniformly bounded degree on the space of rooted graphs, weak convergence implies pointwise convergence of the spectral measures. Our proof relies on their ideas, though we have a different approach, which gives us the possibility to define spectral measures in a more general way. They also mentioned the possibility of applying their result to simplicial complexes by identifying the $p$-simplices with vertices and putting ${{\mathbb Z}}$-labelled edges between them, according to the matrix coefficients of the $p$-th Laplace operator.

\[theorem-approximation\] Let $(\mu_n)_{n\in{{\mathbb N}}}$ be a sequence of sofic random rooted simplicial complexes with 