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Math Help - Example of function not integrable and the square is

  1. #1
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    Example of function not integrable and the square is

    I'm looking for the example of a function f:[0,1]->R that is not integrable on [0,1] but the square of the function is.

    I've tried 1/x and lnx and e^x and a bunch of others, I'm thinking it could have something to do with a transformation of (-x^3) then when it's squared the function is x^6, but I'm not sure if this is the right track. Any ideas?
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  2. #2
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    Re: Example of function not integrable and the square is

    I figured it out. The sin(1/x) integral does not exist and the integral of the square does?
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    Super Member girdav's Avatar
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    Re: Example of function not integrable and the square is

    What does Cauchy-Schwarz inequality give?
    edit : (do you mean Riemann or Lebesgue integrable?)
    Last edited by girdav; July 5th 2011 at 08:11 AM.
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  4. #4
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    Re: Example of function not integrable and the square is

    Quote Originally Posted by CountingPenguins View Post
    I'm looking for the example of a function f:[0,1]->R that is not integrable on [0,1] but the square of the function is.
    What about f(x) = \left\{ {\begin{array}{rl}   {1,} & {x \in \mathbb{Q}}  \\   { - 1,} & {x \notin \mathbb{Q}}  \\ \end{array} } \right.
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Example of function not integrable and the square is

    Quote Originally Posted by Plato View Post
    What about f(x) = \left\{ {\begin{array}{rl}   {1,} & {x \in \mathbb{Q}}  \\   { - 1,} & {x \notin \mathbb{Q}}  \\ \end{array} } \right.
    Proof that f(x) is not not integrable on [0,1] (or on any closed interval [a,b]).



    For every possible partition P=\{\Delta {x_1},\Delta {x_2},...,\Delta {x_n}\} if we choose rational points t_i\in\Delta {x_i}, we'll get:

    Riemann's sum: \sum_{i=1}^{n}f(t_i)\Delta {x_i}=\sum_{i=1}^{n}1\Delta {x_i}=1-0=1

    Now, do the same for k_i irrational.}
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  6. #6
    Senior Member Tinyboss's Avatar
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    Re: Example of function not integrable and the square is

    As girdav noted, it matters whether we're talking Riemann or Lebesgue. The rational/irrational function above is Lebesgue integrable. Think about the harmonic series to find a function not Lebesgue integrable but whose square is.
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  7. #7
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    Re: Example of function not integrable and the square is

    Reimann
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  8. #8
    Super Member girdav's Avatar
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    Re: Example of function not integrable and the square is

    Quote Originally Posted by CountingPenguins View Post
    Riemann
    Hence Plato's answer solves the problem.
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