- #1
ianchenmu
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Homework Statement
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.
Homework Equations
n/a
The Attempt at a Solution
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##.