case, as long as the Laplace operator is self-adjoint, we will assume in the following that $\mu$ is an unimodular random rooted simplicial complex of bounded degree. We denote the projection valued measure of the $p$-th Laplace operator $\Delta_p$ on $C_p^{(2)}({\mathcal{SC}_*},\mu)$ by $E_{\Delta_p}$.

We define the *p-th spectral measure* of an unimodular random rooted simplicial complex of bounded degree $\mu$ as $$\nu_p^\mu(S)= {\operatorname{tr}}_p^\mu(E_{\Delta_p}(S)),$$ for every Borel set $S\subset {{\mathbb R}}$.

As remarked after Proposition \[prop-dykema\], this is equivalent to $$\nu_p^\mu(S)=\int_{{\mathcal{SC}_*}} \nu_p^K(S)d\mu,$$ where $$\nu_p^K(S)=\sum_{\substack{\sigma\in K(p)\\
x\in\sigma}}\frac{\langle E_{\Delta_p}^K(S)\sigma,\sigma\rangle}{p+1}$$ is the spectral measure of the $p$-th Laplace operator of $K$ with $E_{\Delta_p}^K$ the corresponding projection valued measure. This approach was for example chosen in [@abert-thom-virag] and [@linial-peled] in the context of rooted graphs.\
Note that $\nu_p^\mu$ is not a probability measure, but it is finite as long as the expected $p$-degree at the root $\mathbb{E}_\mu(\deg_p)=\int_{{\mathcal{SC}_*}}\deg_p(x)d\mu$ is finite (which is the case if $\mu$ has bounded degree), since we have $$\begin{aligned}
\nu_p^\mu(\mathbb{R})&={\operatorname{tr}}_p^\mu(E_p(\mathbb{R}))={\operatorname{tr}}_p^\mu({\operatorname{Id}}_p)=\mathbb{E}_\mu(\deg_p)/(p+1).\end{aligned}$$

For an unimodular random rooted simplicial complex $\mu$ of bounded degree the $p$-th $\ell^2$-Betti number is $$\begin{aligned}
\beta_p^{(2)}(\mu)=\nu_p^{\mu}(\{0\}).\end{aligned}$$

Benjamini-Schramm convergence and approximation Theorem
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\[definition-BSconvergence\] We say a sequence $K_n$ of simplicial complexes converges *Benjamini-Schramm* to $K$ if $$\mu_{K_n}(U_{[L,y]}(r))\to \mu_{K}(U_{[L,y]}(r))$$ for all 