Integration of differential forms on complex manifolds is defined by using a partition of unity with compact supports subordinated to a covering by holomorphic charts. Surprisingly, this was not necessary in our non-archimedean algebraic setting as we have defined integration by using a single suitable tropical chart. In fact, the use of a smooth partition of unity $(\phi_{j})_{j\in J}$ with compact supports subordinate to an open covering of $W$ by tropical charts $(V_{i},\varphi_{U_{i}})_{i\in I}$ would not work here directly. To illustrate this, suppose that $\alpha\in A^{n,n}_{c}(W)$ is given on $V_{i}$ by $\alpha_{i}\in A^{n,n}({\rm trop}_{U_{i}}(V_{i}))$. If the functions $\phi_{j}$ are of the form $\phi_{j}=f_{j}\circ{\rm trop}_{U_{i(j)}}$ for some $V_{i(j)}\supset{\rm supp}(\phi_{j})$ and $f_{j}\in C^{\infty}_{c}({\rm trop}_{U_{i(j)}}(V_{i(j)}))$, then we could set $\int_{W}\alpha=\sum_{j\in J}\int_{{\rm Trop}(U_{i(j)})}f_{j}\alpha_{i(j)}$. However, the functions $\phi_{j}$ could not be expected to have this form and so this approach fails.

Chambert-Loir and Ducros define integration more generally for differential forms on paracompact good analytic spaces (see [CD12], §3.8). The idea is to use a covering by the interiors of affinoid subdomains. Then there is a smooth partition of unity with supports subordinated to this covering which reduces the problem to defining integration over an affinoid subdomain. But in the affinoid case, one can find a single tropical chart of integration similarly as in Proposition 5.13. It follows from Remark 7.6 and Proposition 7.11 that both definitions give the same integral on the analytification of an algebraic variety.