## Riemann integrable then J-integrable

Let $\displaystyle E\subset\mathbb{R}^n$ be a closed Jordan domain and $\displaystyle f:E\rightarrow\mathbb{R}$ a bounded function. We adopt the convention that $\displaystyle f$ is extended to $\displaystyle \mathbb{R}^n\setminus E$ by $\displaystyle 0$.
Let $\displaystyle \jmath$ be a finite set of Jordan domains in $\displaystyle \mathbb{R}^n$ that cover $\displaystyle E$.

Define $\displaystyle M_J=sup\left \{ f(x)\;|\;x\in J \right \}$, $\displaystyle m_J=inf\left \{ f(x)\;|\;x\in J \right \}$

$\displaystyle W(f;\jmath )=\sum_{J\in\jmath ,J\cap E\neq \varnothing }M_JVol(J)\;\;\;\;\;\;\;\;\;\;$ (upper R-sum)
$\displaystyle w(f;\jmath )=\sum_{J\in\jmath ,J\cap E\neq \varnothing }m_JVol(J)\;\;\;\;\;\;\;\;\;\;$ (lower R-sum)

Define $\displaystyle \overline{vol}(f;E)=inf\left \{ W(f;\jmath ) \right \}\;$, $\displaystyle \;\underline{vol}(f;E)=sup\left \{ w(f;\jmath ) \right \}$.

Say that $\displaystyle f$ is $\displaystyle J$-integrable on $\displaystyle E$ if $\displaystyle \overline{vol}(f;E)=\underline{vol}(f;E)$.

**Prove** that if $\displaystyle f$ is Riemann integrable on $\displaystyle E$ then it is $\displaystyle J$-integrable.

How to relate this? The definition of Riemann integrable has only a difference that $\displaystyle \jmath$ is an n-dimensional rectangle and $\displaystyle J$ is a grid on $\displaystyle \jmath$.