Let $A_{c}(U)$ denote the space of superforms on $U$ with compact support in $U$. For $\alpha\in A_{c}(U)$, we define

$$\int_{U}\alpha:=\int_{U}\alpha_{LL}dx_{1}\wedge\dots\wedge dx_{r}$$

with $L=\{1,\dots,r\}$ and the usual integration of $r$-forms with respect to the orientation induced by the choice of coordinates on the right hand side. If $F$ is an affine map as in 2.3 and if $r=r^{\prime}$, then we have the transformation formula

$$\int_{V}F^{*}(\alpha)=|\det(F)|\int_{U}\alpha$$

(1)

(see [La12], equation (2.3)). We conclude that the definition of the integral depends only on the underlying integral ${\mathbb{R}}$-affine structure of $N_{\mathbb{R}}$.