1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How to prove if Riemann integrable then J-integrable

  1. Mar 31, 2013 #1
    1. The problem statement, all variables and given/known data
    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.





    2. Relevant equations

    n/a

    3. 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##.
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: How to prove if Riemann integrable then J-integrable
Loading...