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Thread: Integrability

  1. #1
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    Integrability

    Give an example of a function $\displaystyle f:\mathbb R \rightarrow \mathbb R$ such that $\displaystyle f$ is integrable but $\displaystyle f^2$ is not integrable. Prove your result.

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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    $\displaystyle f(x)=\left\{\left\begin{array}{ll}1 \ \mbox{if }x \in \mathbb{Q}\\-1 \ \mbox{otherwise.}\end{array}$
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  3. #3
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    There is a problem with the OP.
    There is a theorem (its proof is quite tedious): If $\displaystyle f$ is integrable on $\displaystyle [a,b]$ then $\displaystyle f^2$ is integrable on $\displaystyle [a,b]$.

    In the OP the finite interval was not included. So I am not sure what is meant.

    However, in the example given by Bruno J $\displaystyle f$ is not integrable but $\displaystyle f^2$ is integrable. The inverse of the OP.
    Last edited by Plato; Oct 20th 2010 at 02:27 PM.
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  4. #4
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    This is the definition I know:



    $\displaystyle f$ is integrable if $\displaystyle \int|f|< \infty$. Now in Bruno J's example I don't think either $\displaystyle f$ or $\displaystyle f^2$ are integrable since $\displaystyle |f|$and$\displaystyle |f^2|$ are both identically 1 and the integral is not finite.
    With regards to Plato's post, well it tells us that we can't use a function integrable on [a,b] since then $\displaystyle f^2$ is going to be integrable. But any function defined on a subset of $\displaystyle \mathbb{R}$ should do. So still thinking.....
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  5. #5
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by Plato View Post
    There is a problem with the OP.
    There is a theorem (its proof is quite tedious): If $\displaystyle f$ is integrable on $\displaystyle [a,b]$ then $\displaystyle f^2$ is integrable on $\displaystyle [a,b]$.

    In the OP the finite interval was not included. So I am not sure what is meant.

    However, in the example given by Bruno J $\displaystyle f$ is not integrable but $\displaystyle f^2$ is integrable. The inverse of the OP.
    Oops!
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