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Math Help - Riemann integral is zero for certain sets

  1. #1
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    Riemann integral is zero for certain sets

    The question is:


    Let \pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}. Prove that if E\subset\pi is a closed Jordan domain, and f:E\rightarrow\mathbb{R} is Riemann integrable, then \int_{E}f(x)dV=0.


    (How to relate the condition it's Riemann integrable to the value is 0? The textbook I use define f is integrable on E iff \;\;\;\;(L)\int_{E}fdV=(U)\int_{E}fdV)
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  2. #2
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    Re: Riemann integral is zero for certain sets

    Quote Originally Posted by ianchenmu View Post
    The question is:


    Let \pi=\left \{ x\in\mathbb{R}^n\;|\;x=(x_1,...,x_{n-1}, 0) \right \}. Prove that if E\subset\pi is a closed Jordan domain, and f:E\rightarrow\mathbb{R} is Riemann integrable, then \int_{E}f(x)dV=0.


    (How to relate the condition it's Riemann integrable to the value is 0? The textbook I use define f is integrable on E iff \;\;\;\;(L)\int_{E}fdV=(U)\int_{E}fdV)
    What is the (L) and (U)? In any case, since the integral exists, it must be zero since the set you're integrating over has Lebesgue measure zero.
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  3. #3
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    Re: Riemann integral is zero for certain sets

    Quote Originally Posted by Gusbob View Post
    What is the (L) and (U)? In any case, since the integral exists, it must be zero since the set you're integrating over has Lebesgue measure zero.
    (L)\int_{E}fdV=\underset{G}{sup}L(f;G),

    (U)\int_{E}fdV=\underset{G}{inf}U(f;G),

    U(f;G)=\sum_{R_j\cap E\neq \O }M_j|R_j| where M_j=sup_{x\in R_j}f(x),
    L(f;G)=\sum_{R_j\cap E\neq \O }m_j|R_j| where m_j=inf_{x\in R_j}f(x).

    ( E is an Jordan region in \mathbb{R}^n, f is a bounded function, R is an n-dimensional rectangle s.t. E\subseteq R.
    G=\left \{ R_1,...,R_p \right \} is a grid on R. (Extend f to \mathbb{R}^n by setting f(x)=0 for x\in\mathbb{R}^n\setminus E.))

    Can you use these backgrounds and terminologies to explain? Thanks!
    Last edited by ianchenmu; March 31st 2013 at 02:35 AM.
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