$\mu$ is of *bounded degree* if $\deg(\mu)<\infty$.\
For comparison, the *expected p-degree* of $\mu$ is $$\begin{aligned}
\mathbb{E}(\deg_p)=\int_{{\mathcal{SC}_*}}\deg_p(x)d\mu.\end{aligned}$$

Now we will give some examples of (unimodular) random rooted simplicial complexes:

\[example-measures\]

1.  Every finite connected simplicial complex $K$ defines an unimodular [*rrsc*]{} of bounded degree by: $$\begin{aligned}
    \mu_K:=\sum_{x\in V(K)}\frac{\delta_{[K,x]}}{| V(K)|} ,\end{aligned}$$ where $\delta_{[K,x]}$ denotes the Dirac measure of the point $[K,x]$ in ${\mathcal{SC}_*}$. If $K$ has more than one connected component, we define the associated unimodular [*rrsc*]{} by: $$\begin{aligned}
    \mu_K:=\sum_{x\in V(K)}\frac{\delta_{[K_x,x]}}{| V(K)|}, \end{aligned}$$ with $K_x$ the connected component of $x\in V(K)$.

2.  There is a more general construction, which also applies to infinite complexes. Let $K$ be a connected locally finite simplicial complex with unimodular automorphism group $\mathrm{Aut}(K)$. Further, let $\{x_1,x_2,...\}$ be a complete orbit section of $V(K)$. Then there exist an unimodular probability measure $\mu_K$ on ${\mathcal{SC}_*}$, fully supported on the rooted isomorphism classes of $K$, if $$\begin{aligned}
    c\coloneqq\sum_i \frac{1}{|\mathrm{Stab}(x_i)|}<\infty,\end{aligned}$$ where $|\cdot|$ denotes the Haar measure. The unimodular [*rrsc*]{} is given by: $$\begin{aligned}
    \mu_{K}:=\frac{1}{c}\sum_{i}\frac{\delta_{[K,x_i]}}{|\mathrm{Stab}(x_i)|}.\end{aligned}$$ In [@aldous-lyons Theorem 3.1] Aldous and Lyons give a proof of this statement for graphs, which also applies to simplicial complexes. They proved even more: The measure $\mu_G$ associated with a graph $G$ is unimodular if 