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Math Help - A problem on L^p-space

  1. #1
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    A problem on L^p-space

    A real-valued measurable function f is said to be in L^p-space if |f|^p has finite integral. So if f\in L^p, |f|^p must be Lebesgue integrable. My question is: in this case, is it possible that f itself is not Lebesgue integrable? Under what condition can we have that an f in L^p is also Lebesgue integrable? Thanks!
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  2. #2
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    Hmm. I seem to remember there are theorems about L^{p}\subset L^{q} for certain conditions on p and q. Look in Royden under that subject.
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    I searched chapter 6 in Royden's book, as well as exercises, but failed to find such theorem. Is it located in some other chapter?
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  4. #4
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    Quote Originally Posted by zzzhhh View Post
    A real-valued measurable function f is said to be in L^p-space if |f|^p has finite integral. So if f\in L^p, |f|^p must be Lebesgue integrable. My question is: in this case, is it possible that f itself is not Lebesgue integrable? Under what condition can we have that an f in L^p is also Lebesgue integrable? Thanks!
    You can take a trivial counter-example:
    <br />
\int |\frac{1}{x} \mathbf{1}_{[1, \infty)}|= \infty<br />

    Now take p=2 for a contrast.

    As for L^p \subset L^1, this is true under finite measures (you can use Jensen). I don't know about conditions on general spaces but being compactly supported is obviously enough.
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