The support of a function on $\mathbb{R}^n$ and its compactly supported Riemann integral

Let $n\ge1$. For $f:\mathbb R^n\to\mathbb R$, its support is $\operatorname{supp}f:=\overline{\{x\in\mathbb R^n:f(x)\ne0\}}.$ The function is compactly supported if $\operatorname{supp}f$ is compact.

A compactly supported $f$ is compactly supported Riemann integrable if there is a nondegenerate closed rectangle $Q$ with $\operatorname{supp}f\subseteq\operatorname{int}Q$ such that $f|_Q$ is Riemann integrable. Its integral over Euclidean space is defined by $\int_{\mathbb R^n}f:=\int_Qf.$ The value is independent of $Q$ by The Riemann integral of a compactly supported function is independent of its bounding rectangle ↗. When the support is empty, $f=0$ and the value is $0$.