The Riemann integral of a bounded function over a bounded Jordan measurable set

Let $E\subseteq\mathbb R^m$ be bounded in the metric sense of Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space and Jordan measurable, and let $f:E\to\mathbb R$ be bounded. Choose a nondegenerate rectangle $Q\supseteq E$, whose existence follows from Jordan inner and outer content and Jordan measurable bounded sets in $\mathbb{R}^m$, and define the zero extension $\widetilde f_Q(x):=\begin{cases}f(x),&x\in E,\\0,&x\in Q\setminus E.\end{cases}$ The function $f$ is Riemann integrable over $E$ when $\widetilde f_Q$ is integrable over $Q$, and then $\int_Ef:=\int_Q\widetilde f_Q.$ Independence of the bounding rectangle, for both integrability and value, is proved in The Riemann integral over a Jordan set is independent of the bounding rectangle ↗ and recorded as the definition's forward justification. For $f=1$, the zero extension is $1_E$, so A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content gives $\int_E1=\operatorname{cont}(E)$.