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Math Help - Integrability

  1. #1
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    Integrability

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

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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    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 f is integrable on [a,b] then f^2 is integrable on [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 f is not integrable but f^2 is integrable. The inverse of the OP.
    Last edited by Plato; October 20th 2010 at 03:27 PM.
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  4. #4
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    This is the definition I know:



    f is integrable if \int|f|< \infty. Now in Bruno J's example I don't think either f or f^2 are integrable since |f|and |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 f^2 is going to be integrable. But any function defined on a subset of \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 f is integrable on [a,b] then f^2 is integrable on [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 f is not integrable but f^2 is integrable. The inverse of the OP.
    Oops!
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