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Thread: Are the following Lebesgue integrable?

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    Are the following Lebesgue integrable?

    (You can use standard results in any proofs). Are the following 2 functions Lebesgue integrable over the interval I?

    A) I = [1,\infty) with f(x) = (-1)^n/n if n \leq x < n+1, n = 1,2,3...

    B) I = [0,1] with f(x) = (-1)^{n(x)}/n(x) where n(x) is the largest integer n such that nx \leq 1
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  2. #2
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    Quote Originally Posted by Amanda1990 View Post
    (You can use standard results in any proofs). Are the following 2 functions Lebesgue integrable over the interval I?

    A) I = [1,\infty) with f(x) = (-1)^n/n if n \leq x < n+1, n = 1,2,3...

    B) I = [0,1] with f(x) = (-1)^{n(x)}/n(x) where n(x) is the largest integer n such that nx \leq 1
    For A), remember that the Lebesgue integral is an absolute integral. In other words, if f is integrable then so is |f|.

    For B), show that f is measurable. Since it is bounded, on a bounded interval, it must then be integrable.
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    Thanks for you help. However for B) I've got that f is not integrable. Surely f is unbounded, as 1/n(x) can become arbitrarily large for small x?
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    Quote Originally Posted by Amanda1990 View Post
    Thanks for you help. However for B) I've got that f is not integrable. Surely f is unbounded, as 1/n(x) can become arbitrarily large for small x?
    If x is small then n(x) will be large, so 1/n(x) will be small.
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    Ouch. Good point...
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