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.