Consequently, if the image of a complex contains $\Upsilon(k)$, then it is already in the image of the $(k_0+1)$-ball and hence the preimage consists of all $k_0+1$-neighbourhoods of balls with $\Upsilon(k)$ in their image i.e. $$\Psi^{-1}([k])=\bigcup_{\substack{[L,y]\in \mathcal{B}(k_0+1) \\
\Upsilon(k)\in \Psi ([L,y])}} U_{k_0+1}(L,y).$$ This is a countable union of open sets, therefore $\Psi$ is continuous.

Let $\mu$ be a random rooted simplicial complex. We define the $p$-th *simplicial $\ell^2$-chain module of ${\mathcal{SC}_*}$ with respect to $\mu$* as the direct integral Hilbert space $$C_{p}^{(2)}({\mathcal{SC}_*},\mu)\coloneqq\int_{{\mathcal{SC}_*}}^\oplus C_p^{(2)}((K,x)_\mathbb{N})d\mu([K,x]),$$ where $C_p^{(2)}((K,x)_\mathbb{N})$ denotes the space of complex or real valued square-summable simplicial $p$-chains of the simplicial complex $(K,x)_\mathbb{N}$.

The space $C_p^{(2)}({\mathcal{SC}_*},\mu)$ consists of $\mu$-equivalence classes of $\mu$-measurable functions $f$ on ${\mathcal{SC}_*}$ such that $f([K,x])\in C^{(2)}_p((K,x)_\mathbb{N})$ almost everywhere and $\|f\|\in L^2({\mathcal{SC}_*},\mu)$. We call a function $f$ *measurable* if $$[K,x]\mapsto \langle f([K,x]), \delta_\sigma ([K,x]) \rangle$$ is measurable for all oriented $p$-simplices $\sigma$ of ${\Delta(\mathbb{N}_0)}(p)$, where $$\begin{aligned}
\delta_\sigma([K,x])=\left\{\begin{array}{ll}
\sigma & \text{if }\sigma\in (K,x)_{{\mathbb N}}\\
0 & \text{otherwise.}
\end{array}\right.\end{aligned}$$ The set $\{\delta_\sigma\;|\; \sigma \in {\Delta(\mathbb{N}_0)}(p)\}$ is a so called *fundamental sequence for* $C_p^{(2)}({\mathcal{SC}_*},\mu)$. The inner product is given by $$\langle f,g\rangle_\mu =\int_{{\mathcal{SC}_*}} \langle f([K,x]),g([K,x])\rangle_{(K,x)_{{\mathbb N}}} d\mu([K,x]).$$

Let $T_{[K,x]}$ be a continuous linear mapping of $C_p^{(2)}((K,x)_{{\mathbb N}})$ to $C_q^{(2)}((K,x)_{{\mathbb N}})$ for every 