Zero Extension Continuity and Box Independence of the Iterated Integral

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 $\Omega$, with coefficient function $f$ and zero extension $\tilde f$, all in the sense of that definition. Then the following hold.

1. The function $\tilde f:D\to\mathbb{R}$ is continuous at every point of $D$.

2. There exists a closed box $B\subseteq D$ with $\operatorname{supp}\omega\subseteq B$. When $D=H^n$ is the closed upper half-space, such a box has the form $B=[a_1,b_1]\times\cdots\times[a_n,b_n]$ with $a_n\ge 0$.

3. Let $B=[a_1,b_1]\times\cdots\times[a_n,b_n]\subseteq D$ be any closed box with $\operatorname{supp}\omega\subseteq B$, where $a_j,b_j\in\mathbb{R}$ with $a_j\le b_j$ for each $j\in\{1,\dots,n\}$. For fixed $(t_2,\dots,t_n)\in[a_2,b_2]\times\cdots\times[a_n,b_n]$ define

$$G_1(t_2,\dots,t_n)=\int_{a_1}^{b_1}\tilde f(t_1,t_2,\dots,t_n)\,dt_1,$$

as a one-dimensional Riemann integral; recursively, for $r\in\{2,\dots,n\}$ and fixed $(t_{r+1},\dots,t_n)\in[a_{r+1},b_{r+1}]\times\cdots\times[a_n,b_n]$ define

$$G_r(t_{r+1},\dots,t_n)=\int_{a_r}^{b_r}G_{r-1}(t_r,\dots,t_n)\,dt_r.$$

Then at every stage the integrand is a continuous function of the integration variable on the corresponding closed interval, hence Riemann integrable, each function $G_r$ is continuous in its remaining variables, and the final value $G_n\in\mathbb{R}$ is well defined.

4. The value $G_n$ obtained in claim 3 is the same for every closed box $B\subseteq D$ with $\operatorname{supp}\omega\subseteq B$.