$[K,x]\in{\mathcal{SC}_*}$. We call such a field of linear maps $[K,x]\mapsto T_{[K,x]}$ *measurable* if $$[K,x]\mapsto \langle T_{[K,x]}\delta_\sigma,\delta_\theta \rangle$$ is measurable for every $\sigma\in {\Delta(\mathbb{N}_0)}(p)$ and $\theta\in{\Delta(\mathbb{N}_0)}(q)$.

We call an operator $$T\colon C_p^{(2)}({\mathcal{SC}_*},\mu)\to C_q^{(2)}({\mathcal{SC}_*},\mu)$$ *decomposable* if there is a measurable field $[K,x]\to T_{[K,x]}$ of operators, such that for every element $f\in C_p^{(2)}({\mathcal{SC}_*},\mu)$ in the domain of $T$ and every $[K,x]\in {\mathcal{SC}_*}$, $f([K,x])$ is in the domain of $T_{[K,x]}$, $[K,x]\mapsto \|T_{[K,x]}f([K,x])\|$ is square integrable and $$(Tf)([K,x])=T_{[K,x]}f([K,x])$$

for $\mu$ almost all $[K,x]\in{\mathcal{SC}_*}$.\
$T$ is closed (resp. densely defined, self-adjoint) if and only $T_{[K,x]}$ is closed (resp densely defined, self-adjoint) $\mu$ almost everywhere [@nussbaum Theorem 3].

[@dixmier II.3] Let $[K,x]\to T_{[K,x]}$ be a measurable field of operators on ${\mathcal{SC}_*}$. If the $\mu$-essential supremum $D$ of $[K,x]\mapsto \|T_{[K,x]}\|$ is finite, we call $[K,x]\to T_{[K,x]}$ *$\mu$-essentially bounded* and the field defines a bounded operator $T$ on $C_p^{(2)}({\mathcal{SC}_*},\mu)$ with $$\|T\|=D.$$

For more details about direct integrals see [@dixmier] or [@nielsen]. Suppose we have a Borel probability measure $\mu$ such that the fields of boundary operators $$\partial_p\colon [K,x]\mapsto [\partial_{p,[K,x]}\colon C_p^{(2)}((K,x)_\mathbb{N})\to C_{p-1}^{(2)}((K,x)_\mathbb{N})]$$ and its adjoints $$\partial_p^*\colon [K,x]\mapsto [\partial_{p,[K,x]}^*\colon C_{p-1}^{(2)}((K,x)_\mathbb{N})\to C_{p}^{(2)}((K,x)_\mathbb{N})]$$ are $\mu$-essentially bounded. Therefore, the measurable fields of operators define bounded linear mappings between direct integral Hilbert spaces $$\begin{aligned}
\int^\oplus_{{\mathcal{SC}_*}}\partial_{p,[K,x]} 