Integral of a Compactly Supported Continuous n-Form on a Euclidean or Half-Space Domain

Let $n\in$ $\mathbb{N}$, let $\Omega$ be an admissible domain in Euclidean space $\mathbb{R}^n$ with ambient set $D$ in the sense of Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain, and let $\omega$ be a continuous differential $n$-form on $\Omega$ that is compactly supported in $\Omega$ in the sense of that definition.

By Zero Extension Continuity and Box Independence of the Iterated Integral there exists a closed box $B\subseteq D$ containing $\operatorname{supp}\omega$, the iterated one-dimensional Riemann integrals of the zero extension $\tilde f$ of the coefficient function of $\omega$ over any such box exist, and the resulting value is independent of the chosen box. The integral of $\omega$ over $\Omega$, denoted

$$\int_{\Omega}\omega,$$

is defined to be this common value.