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Thread: Lebesgue Integral

  1. #1
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    Lebesgue Integral

    hi, i need help for this question:
    $\displaystyle {f}_{n}(x)=n*{\varphi }_{[0,1/n]} $ where $\displaystyle {\varphi}_{[0,1/n]}$ is characterestic function.

    1-)what is the value of $\displaystyle \int {f}_{n} d\mu $ over the interval [0,1] ?
    2-) if n goes to infinity and $\displaystyle {f}_{n}(x)\to f(x)$ , what is the value of $\displaystyle \int f d\mu $ over the interval [0,1]

    thanks for any help.
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  2. #2
    Super Member girdav's Avatar
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    1) What is the integral of a characteristic function ?
    2) Try to find $\displaystyle f$.
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  3. #3
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    1-)well the function $\displaystyle {f}_{n} (x)$ will be equal to $\displaystyle n$ for $\displaystyle x \epsilon [0,1/n] $ and otherwise it is 0.
    so i find that $\displaystyle \int {f}_{n } d\mu = \int n d\mu $ over [0,1]. $\displaystyle \int nd\mu =n\int d\mu =n $ over [0,1].

    is that right?
    2-) i can't figure how to find f. i sense f(x) must be zero when i graph $\displaystyle {f}_{n}$
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  4. #4
    Super Member girdav's Avatar
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    1) No, we have $\displaystyle \int f_nd\mu =n\mu ([0,\frac 1n])=1$.
    2) $\displaystyle f=0$ almost everywhere, so what about $\displaystyle \int fd\mu$ ?
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  5. #5
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    1-) i got it now when graphing. i used the measure of [0,1].
    2-) it equals 0
    thanks anyway.
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