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Math Help - finding sequence of functions! difficult problem about integration..

  1. #1
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    finding sequence of functions! difficult problem about integration..

    Let f,g are ANY non-negative and measurable. \mu is measure
    Find two function $f_n,g_n$ such that HAVE ONLY FINITELY MANY VALUE and
    for any natural number $n$,
    |\int^{\infty}_0 \mu(\{ x : f(x) > t \}) dt - \int^{\infty}_0 \mu(\{ x : f_n(x) > t \}) | \leq \frac {C}{n} and
    |\int^{\infty}_0 \mu(\{ x : g(x) > t \}) dt - \int^{\infty}_0 \mu(\{ x : g_n(x) > t \}) | \leq \frac {C}{n} and
    |\int^{\infty}_0 \mu(\{ x : f(x) + g(x) > t \}) dt - \int^{\infty}_0 \mu(\{ x : f_n(x) + g_n(x) > t \}) | \leq \frac {C}{n}.
    And C is independent of n.

    *Important* above integral is (improper) riemann integral !!
    (In fact, this problem is from ANALYSIS by Lieb and Loss.. so definition of integral is different from other books
    it defines integral by \int f d\mu : = \int^{\infty}_0 \mu({x: f(x) > t}) dt so i wrote the problem as above..)
    P.S. sorry for poor english..

    -I correct some mistakes -
    Last edited by ramsey88; February 15th 2009 at 04:46 AM. Reason: mistakes
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  2. #2
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    Quote Originally Posted by ramsey88 View Post
    Let f,g are non-negative and measurable. \mu is measure
    Find two function f_n,g_n such that HAVE ONLY FINITELY MANY VALUE and
    for any natural number n,
    |\int^{\infty}_0 \mu(\{ x : f(x) > t \}) dt - \int^{\infty}_0 \mu(\{ x : f_n(x) > t \}) | \leq \frac {C}{n} and
    |\int^{\infty}_0 \mu(\{ x : g(x) > t \}) dt - \int^{\infty}_0 \mu(\{ x : g_n(x) > t \}) | \leq \frac {C}{n} and
    |\int^{\infty}_0 \mu(\{ x : f(x) + g(x) > t \}) dt - \int^{\infty}_0 \mu(\{ x : f_n(x) + g(x) > t \}) | \leq \frac {C}{n}.
    And C is independent of n.

    *Important* above integral is (improper) riemann integral !!
    (In fact, this problem is from ANALYSIS by Lieb and Loss.. so definition of integral is different from other books
    it defines integral by \int f d\mu : = \int^{\infty}_0 \mu({x: f(x) > t}) dt so i wrote the problem as above..)
    P.S. sorry for poor english..
    I'm not clear on what the problem requires. Is there any requirement that would prevent a trivial example like f_n(1)= 1 and undefined for other x, for all n, g_n(0)= 1 and undefined for other x, for all n?

    In that case, all integrals are 0 so you can take C to be any positive number.
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    I'm not clear on what the problem requires. Is there any requirement that would prevent a trivial example like f_n(1)= 1 and undefined for other x, for all n, g_n(0)= 1 and undefined for other x, for all n?

    In that case, all integrals are 0 so you can take C to be any positive number.
     f_n, g_n should satisfy above condition for any non-negative,measurable function f,g.
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