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Math Help - pointwise convergence

  1. #1
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    Cool pointwise convergence

    I need help with this problem.

    Give an example of a sequence of integrable functions fn:[a,b]-->R convergent (pointwise) to a non-integrable function f:[a,b]-->R.
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  2. #2
    Senior Member Tinyboss's Avatar
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    What kind of integration? Riemann, Lebesgue, something else?

    For Riemann integration, enumerate the rationals in the interval as \{q_i\}_{i=1}^\infty, then let

    f_n(x)=\begin{cases}1&x\in\{q_1,\dots,q_n\}\\0&\te  xt{otherwise.}\end{cases}

    Then \int f_n=0 for every finite n, but the limit function is not integrable.
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  3. #3
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    Quote Originally Posted by derek walcott View Post
    I need help with this problem.

    Give an example of a sequence of integrable functions fn:[a,b]-->R convergent (pointwise) to a non-integrable function f:[a,b]-->R.
    For any integral: choose a non-integrable nonnegative function f:[a,b]\to\mathbb{R}_+ (any one you want), then f_n=\min(f,n) is bounded by n, hence integrable on [a,b], and converges pointwise to f.
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  4. #4
    Super Member Failure's Avatar
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    Quote Originally Posted by Laurent View Post
    For any integral: choose a non-integrable nonnegative function f:[a,b]\to\mathbb{R}_+ (any one you want), then f_n=\min(f,n) is bounded by n, hence integrable on [a,b], and converges pointwise to f.
    I don't think that from the assumption that f be non-negative, non-integrable it follows that f_n =\min(f,n) is integrable.
    For the Riemann integral it is easy to give an example of a non-negative and bounded function that is non-integrable (e.g. the characteristic function of \mathbb{Q}), and in the case of the Lebesgue integral we must consider the possibility that f might not even be measurable.

    So, I think, one would have to be a bit more specific about the cause of the non-integrability of f.
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  5. #5
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    Quote Originally Posted by Failure View Post
    I don't think that from the assumption that f be non-negative, non-integrable it follows that f_n =\min(f,n) is integrable.
    For the Riemann integral it is easy to give an example of a non-negative and bounded function that is non-integrable (e.g. the characteristic function of \mathbb{Q}), and in the case of the Lebesgue integral we must consider the possibility that f might not even be measurable.

    So, I think, one would have to be a bit more specific about the cause of the non-integrability of f.
    Right, I meant: "choose a measurable non-integrable function f". Since f is the limit of f_n, it has to be measurable.
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