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Math Help - Lebesgue Integral

  1. #1
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    Lebesgue Integral

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

    1-)what is the value of  \int {f}_{n} d\mu over the interval [0,1] ?
    2-) if n goes to infinity and {f}_{n}(x)\to f(x) , what is the value of  \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 f.
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  3. #3
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    1-)well the function {f}_{n} (x) will be equal to n for x \epsilon [0,1/n] and otherwise it is 0.
    so i find that \int {f}_{n } d\mu = \int n d\mu over [0,1]. \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 {f}_{n}
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  4. #4
    Super Member girdav's Avatar
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    1) No, we have \int f_nd\mu  =n\mu ([0,\frac 1n])=1.
    2) f=0 almost everywhere, so what about \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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