Example \[example-measures\].1. to a random sample.

The $\ell^2$-chain complex of ${\mathcal{SC}_*}$
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In this section we will define the simplicial $\ell^2$-chain complex of ${\mathcal{SC}_*}$. For this, we need to choose a representative for each isomorphism class in a measurable way. This was already done by Aldous and Lyons [@aldous-lyons] for rooted networks. Our construction is based on their ideas. We denote the infinite dimensional simplicial complex consisting of all finite subsets of $\mathbb{N}_0$ with $\Delta (\mathbb{N}_0)$. Considered as set, this is just $\mathcal{P}_{<\infty}(\mathbb{N}_0)$.

Since the space of finite subsets of $\mathbb{N}_0$ is countable, every simplex of $\Delta(\mathbb{N}_0)$ can be identified with a natural number. We do this in the following way (diagonal argument): $$\begin{aligned}
\left(\begin{array}{cccccccccc}
{{\mathbb N}}_0& 0&1&2&3&4&5&6&7&... \\
{\Delta(\mathbb{N}_0)}&  \{0\}&\{1\}&\{0,1\}&\{2\}&\{0,2\}&\{1,2\}&\{0,1,2\}&\{3\}&...
\end{array}\right).\end{aligned}$$ and denote this map by $\Upsilon\colon {{\mathbb N}}_0\to {\Delta(\mathbb{N}_0)}$. Subsequently all subsets of $\{0,1,2,...,n\}$ show up in the first $2^{n+1}$ entries. Hence, we can identify *the space of locally finite connected subcomplexes of* $\Delta(\mathbb{N}_0)$ with a subset of the space of sequences $\{0,1\}^{\mathbb{N}_0}$, where $0$ at the n-th position means that the simplex $\Upsilon(n)$ of ${\Delta(\mathbb{N}_0)}$ is not contained in the subcomplex. We give the space of *locally finite connected simplicial subcomplexes of* $\Delta(\mathbb{N}_0)$ the subspace topology with 