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Math Help - Rudin Analysis book:shrink nicely

  1. #1
    Senior Member Shanks's Avatar
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    Rudin Analysis book:shrink nicely

    Let m be the lebesgue measure on R^k, a signed measure u is absolutely continous with respect to m, f \in L^1(R^k) be the Radon-Nikodym derivative of u with respect to m.
    x\in R^k, and for any set sequence \{E_i\} which shrink nicely to x, we have
    \limsup_{i\to \infty}\frac{u(E_i)}{m(E_i)}\leq f(x).
    Prove that -f is the Radon-Nikodym derivative of -u with respect to m.
    and show that \liminf_{i\to \infty}\frac{u(E_i)}{m(E_i)}\geq f(x).
    The Definition of "Shrink nicely" in Rudin's Real and Complex Analysis Book:
    A sequence of measureable sets {E_i} shrink nicely to x if there is a positive number a and a sequence of Ball B(x,r_i) centered at x with radias r_i satisfying
    (1) E_i is contained in B(x,r_i)
    (2) \lim_{i\to \infty}r_i=0
    (3) \left|u\right|(E_i)\geq am(B(x,r_i)), for all i=1,2,....

    I have proved the first problem.
    for the second problem, Completely don't know where to start, Please help me out. Appreciation more than I can say!
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  2. #2
    Moo
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    Hello,

    Fatou's lemma ?
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  3. #3
    Senior Member Shanks's Avatar
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    Topic about differnetiation of measure in real analysis.
    the "shink nicely" section in Rudin' Book "Real and Complex Analysis" in Chpter 7.
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  4. #4
    Senior Member Shanks's Avatar
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    Solved. never mind, forget it!
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