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Math Help - Is this function (Lebesgue) Integrable?

  1. #1
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    Is this function (Lebesgue) Integrable?

    Let f be a Lebesgue integrable function that's equal to 0 for all x not in [a,b]

    Let g be a function that is equal to 0 for all x not in a interval [-d,d] and suppose that g is C^

    Is the function h(x) = f(x)*g(x) integrable?
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  2. #2
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    Quote Originally Posted by southprkfan1 View Post
    Let f be a Lebesgue integrable function that's equal to 0 for all x not in [a,b]

    Let g be a function that is equal to 0 for all x not in a interval [-d,d] and suppose that g is C^

    Is the function h(x) = f(x)*g(x) integrable?
    Your function g is bounded (continuous on a segment), hence fg is integrable.
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  3. #3
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    Quote Originally Posted by Laurent View Post
    Your function g is bounded (continuous on a segment), hence fg is integrable.
    I can "see" how that would follow, but I'm not entirely sure the formal proof.

    EDIT: Would it be that:

     \int gf < \int Kf < \infty where K is the upper bound of g?
    Last edited by southprkfan1; March 27th 2010 at 02:24 PM.
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  4. #4
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    Quote Originally Posted by southprkfan1 View Post
    I can "see" how that would follow, but I'm not entirely sure the formal proof.

    EDIT: Would it be that:

     \int gf < \int Kf < \infty where K is the upper bound of g?
    No, it would be \int |gf|\leq \int K|f|=K\int |f|<\infty
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