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Math Help - Please Look: lebesgue Integration

  1. #1
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    Please Look: lebesgue Integration

    Hi

    I have been working with the topic Lebesgue integration for some time, but I have some questions.

    Lets say the the function f(x) = sin^{-\beta}(x) where \beta \in \mathbb{R} and  x \in ]0, \frac{\pi}{2}[

    I will like to show for which values of \beta that this function f(x) is integratable in 0 and \frac{\pi}{2} and finally can be integrated over the ]0, \frac{\pi}{2}[

    I have a feeling that first I first need to show that f(x) is a messurable function which is also know as a Borel function. I know that my function is mesurable if its continous. Since the preimage of f(x) is open, then this is obviously continous, and therefore a Borel function.

    According to the definion I know of Local Lebesgue integratable functions, then if

    f: B \rightarrow \mathbb{C} is called Lebesgue Integratable if f|B \in \mathcal{L}(K) for every compact subset subset K \subseteq B

    Thus \int_{K} |f(x)| dx < \inftyfor every compact subset K \subseteq B

    Could someone here please give me a hint how I by applying this definition Lebesgue integratable function on my function can find the desired values of \beta??

    best Regards
    Billy
    Last edited by Billy2007; November 22nd 2006 at 11:19 AM.
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