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Math Help - Dense subset

  1. #1
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    Cool Dense subset

    Let A be a dense of [a,b] and let f:[a,b]-->R be an integrable function such that f(x)>=0 for every element of x in A. Prove that the (integral of a to b) f >= 0.

    Please help. I have ideas for this but need help solving
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  2. #2
    Senior Member Tinyboss's Avatar
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    I assume you mean f is Riemann-integrable--it isn't true for Lebesgue integration.

    For any partition P, every interval will contain an element of A, and therefore the upper sum is non-negative for that partition. Since f is integrable, the upper sums converge to the integral as the mesh goes to zero, and so the integral is non-negative.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by derek walcott View Post
    Let A be a dense of [a,b] and let f:[a,b]-->R be an integrable function such that f(x)>=0 for every element of x in A. Prove that the (integral of a to b) f >= 0.

    Please help. I have ideas for this but need help solving
    Quote Originally Posted by Tinyboss View Post
    I assume you mean f is Riemann-integrable--it isn't true for Lebesgue integration.

    For any partition P, every interval will contain an element of A, and therefore the upper sum is non-negative for that partition. Since f is integrable, the upper sums converge to the integral as the mesh goes to zero, and so the integral is non-negative.
    Another way to say this depending your def. of Riemann integrable. As Tinyboss said \sup_{x\in I\subseteq[a,b]}f(x)\geqslant 0. Thus, U(P,f)\sum_{j=1}^{n}\sup_{x\in[x_{j-1},x_j]}f(x)\Delta x_j\geqslant 0 regardless of P. Clearly then 0\geqslant U(P,f) for every partition P in \mathcal{P}[a,b]. Thus, \int_a^bf=\inf_{P\in\mathcal{P}[a,b]}U(P,f)\geqslant 0
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