Riemann integrable then J-integrable

[i_a_n]

Let $E\subset\mathbb{R}^n$ be a closed Jordan domain and $f:E\rightarrow\mathbb{R}$ a bounded function. We adopt the convention that $f$ is extended to $\mathbb{R}^n\setminus E$ by $0$.
Let $\jmath$ be a finite set of Jordan domains in $\mathbb{R}^n$ that cover $E$. Define $M_J=sup\left \{ f(x)\;|\;x\in J \right \}$, $m_J=inf\left \{ f(x)\;|\;x\in J \right \}$

$W(f;\jmath )=\sum_{J\in\jmath ,J\cap E\neq \varnothing }M_JVol(J)\;\;\;\;\;\;\;\;\;\;$(upper R-sum)
$w(f;\jmath )=\sum_{J\in\jmath ,J\cap E\neq \varnothing }m_JVol(J)\;\;\;\;\;\;\;\;\;\;$(lower R-sum)Define $\overline{vol}(f;E)=inf\left \{ W(f;\jmath ) \right \}\;$, $\;\underline{vol}(f;E)=sup\left \{ w(f;\jmath ) \right \}$.Say that $f$ is $J$-integrable on $E$ if $\overline{vol}(f;E)=\underline{vol}(f;E)$. **Prove** that if $f$ is Riemann integrable on $E$ then it is $J$-integrable.

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

 

[Jhoneine]

Since $f$ is Riemann-integrable, then by definition, there exists a sequence of partitions $\left \{ P_k \right \}_{k\in\mathbb{N}}$ such that $\lim_{k\rightarrow\infty}R(f;P_k)=\int_E f(x)dx$ and $\lim_{k\rightarrow\infty}L(f;P_k)=\int_E f(x)dx$. Let $\jmath_k$ be the set of $J$-domains that corresponds to the partition $P_k$, so that for each element $x\in P_k$ there is a unique $J\in\jmath_k$ such that $x\in J$. Then we have $$\lim_{k\rightarrow\infty}W(f; \jmath_k)=\lim_{k\rightarrow\infty}R(f;P_k)=\int_E f(x)dx$$and $$\lim_{k\rightarrow\infty}w(f; \jmath_k)=\lim_{k\rightarrow\infty}L(f;P_k)=\int_E f(x)dx$$Hence, we have $$\overline{vol}(f;E)=\lim_{k\rightarrow\infty}W(f; \jmath_k)=\lim_{k\rightarrow\infty}w(f; \jmath_k)=\underline{vol}(f;E)$$Therefore, $f$ is $J$-integrable on $E$.