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Math Help - Prove or disprove integration proof

  1. #1
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    Prove or disprove integration proof

    May I plz get some assistance on the following question;

    If f and g are integrable on a closed set and f(x)<=h(x)<=g(x) for all x in in that set then h (a function) is integrable on that set.
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  2. #2
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    Quote Originally Posted by nankor View Post
    If f and g are integrable on a closed set and f(x)<=h(x)<=g(x) for all x in in that set then h (a function) is integrable on that set.
    What do you mean by "f and g are integrable"?
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  3. #3
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    The statement is false.

    Take f(x)=0 and g(x)=1 on the interval [0,1] and h(x)=1 if x\in\mathbb{Q} and 0 other wise. This shows that the statement is false for the Riemann integral.

    For Lebesgue integrable take h(x)=1 on a set \mu\subset[0,1] and \mu is not Lebesgue measurable, and 0 else where. Then h(x) is not Lebesgue integrable.
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