Choose a common determination of the forms $\theta_{i}$, and define $\int_{X}f\,(\theta_{1}\wedge\dots\wedge\theta_{n})$ using (2.2). The fact that $\int_{X}f\,(\theta_{1}\wedge\dots\wedge\theta_{n})$ does not depend on the choice of a determination $\mathcal{X}$ is a consequence of the projection formula

if $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$, and $D$ is any vertical divisor in $\mathcal{X}^{\prime}$.

Then by construction $\theta_{1}\wedge\dots\wedge\theta_{n}$ can be identified with the atomic measure $\sum_{i}w_{i}\delta_{x_{i}}$ with $w_{i}=(\theta_{1,\mathcal{X}}|_{E_{i}}\cdot\ldots\cdot\theta_{n,\mathcal{X}}|_{E_{i}})$. This measure is supported on the divisorial points associated to the irreducible components of $\mathcal{X}_{0}$. The last statement is clear. ∎